Research Article Mechanical State Model of Multispan Continuous Deep Beam under Concentrated Load

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1 Mathematica Probems i Egieerig Voume 23, Artice ID 66492, 9 ages htt://dxdoiorg/55/23/66492 Research Artice Mechaica State Mode of Mutisa Cotiuous Dee Beam uder Cocetrated Load Chu-Liag Li ad Dog-Miao Zhag Schoo of Commuicatio Sciece & Egieerig, Jii Jiazhu Uiversity, Chagchu 38, Chia Corresodece shoud be addressed to Chu-Liag Li; ichi33@63com Received 7 May 23; Revised 9 November 23; Acceted 9 November 23 Academic Editor: Ashraf M Zekour Coyright 23 C-L Li ad D-M Zhag This is a oe access artice distributed uder the Creative Commos Attributio Licese, which ermits urestricted use, distributio, ad reroductio i ay medium, rovided the origia work is roery cited I the reset study, the mechaica mode of the mutisa cotiuous dee beam uder cocetrated oad was estabished o the state sace theory, ad with the mode, the two-sa cotiuous dee beam was cacuated Ad the cacuatio resuts are cosistet with resuts, which show that the mode without the ae sectio assumtio is accurate ad aicabe The chages of the defectio ad the sectio strai aog the beam height were aayzed with the beam height variatio The resuts showed that the orma stress ad the shear stress coexisted i the cross-sectio of muti-sa cotiuous dee beam, which meas that the shear stress ca cause the cross sectio warig, whie the orma stress ca resut i the sectio i arae with the eutra ayer extrusio I additio, we icidetay foud that, with the icrease of the beam height, the imact of the buckig deformatio ad the extrusio deformatio imact o the beam bedig icrease graduay A ew desig method, which has bee rovided by the estabished mechaica mode of the muti-sa cotiuous dee beam, has exteded the sace of the cacuatio method o traditioa cotiuous dee beam Itroductio The mutisa cotiuous dee beam is maiy used to bear theverticaoad,adtherearetheormastressadtheshear stress i the cross-sectio of the structure whe the dee beam bears cocetrated oad, whie the shear stress ca ead to the cross-sectio warig ad the orma stress ca ead to extrusio i the sectio i arae with the eutra ayer [] With the icrease of the beam height, the shear deformatio ad the extrusio deformatio icreasigy imact o the beam bedig, but the ifuece of the extrusio deformatio is far ess tha that of the shear deformatio Timosheko theory aso adoted the ae sectio assumtio to study the trasverse shear deformatio of thedeebeam,aditermsofthetwofactorsofshear deformatio ad rotary iertia of the structure, it oited out that the rotatio of the cross-sectio was caused by the bedig ad the trasverse shear deformatio This method either ca cacuate the exact shearig strai or satisfy the oshear boudary coditios o the to ad bottom surface,atthesametime[2 4] Leviso [5, 6] ut forward the higher-order shear deformatio theory without the ae sectio assumtio It satisfied the oshear boudary coditios o the uer ad bottom surface But the mode was ot from variatioa ricie because the goverig differetia equatios were estabished with the sectio equiibrium coditio of the beam Gao Wag [7] exoited the geera soutio of Pakovich-Neuber ad the Lure oerator to get directy various oe-dimesioa equatios, which costitute the refied theory of symmetry deformatio beam Meawhie, the fiite eemet method, the fiite itegratio method, ad the series method were aso brought i the fied of mechaics about the dee beam [8 2] The trasverse bedig stress of the dee beam uder uiform oad had bee widey studied i the iterature [3] The mechaica characteristics of the dee beam uder cocetrated oad aso had bee studied i the coditio of arge deformatio [4] The eastic ae robems had bee soved by the sum fuctio method, ad the stress state of the simy suorted dee beam had bee aayzed uder triaguar oad At the same time, the Fourier series soutios were

2 2 Mathematica Probems i Egieerig a b c The first sa The Nth sa Figure : Mesh geeratio of the mutisa cotiuous dee beam The first storey beam The rth storey beam The kth storey beam σ z τ xz F F i reseted i the iterature [5] The aforemetioed theories had cosidered the ifuece of the shear deformatio ad the extrusio deformatio o the dee beam, but they a made differet assumtios to study the distributio of two kids of deformatio I order to get more accurate cacuatio, we coud ot adot the ae sectio assumtio ad the ogitudia fibers oextrusio assumtio to study the deformatio of the mutisa cotiuous dee beam uder cocetrated oad [6], because the distributio of itera force is comicated graduay with the icrease of the degree of statica idetermiacy, eseciay i mutisa cotiuous dee beams Accordig to the iterature, so far, there are few studies about the mutisa cotiuous dee beam uder cocetrated oad; i additio, there is o exaatio about the desig of the mutisa cotiuous dee beam uder cocetrated oad i the curret Cocrete Structure Desig Code, eve ot may studies o the stress distributio aw Thatistosaythereisofeasibeadreasoabecacuatio method o the structure, so it has ractica sigificace to study the mechaica characteristics ad cacuatio method of the mutisa cotiuous dee beam uder cocetrated oad, ad aso it has theoretica vaue to master the stress distributio aw of the dee beam This study aayzed the mutisa cotiuous dee beam uder cocetrated oad without the ae sectio assumtio ad cosequety decreased the mutivariate cacuatio scaeeffectivey;itwioeuaewwaytothemutisa cotiuous dee beam study area z o Figure 2: Forced state of disassembed beam σ x τ xz A σ z A σ z + σ z z dz τ xz + τ xz z dz σ x + σ x x dx Figure 3: Forced state diagram of the rth storey beam removed, ad the equivaet vertica forces were aied o the ocatios of the midde suorts to make sure the ocatios of the midde suorts have o vertica movemet asshowifigure2 23 State Mode Estabishmet of Each Storey Beam Takig ay storey beam r from the first storey to the kth storey, we estabished a oca coordiate system to study the stress state of ay oit A(x, z) i the rth storey beam as show i Figure 3 () Baace equatio of the rth storey beam: x 2 Mechaics Equatio of the Mutisa Cotiuous Dee Beam uder the Cocetrated Load 2 Mesh Geeratio of the Mutisa Cotiuous Dee Beam Itermsofhighdeebeamadtheargeshearstressithe cross-sectio of the dee beam, the mutisa cotiuous dee beam was divided ito some thi storey beams accordig to the materia comositio ad the cacuatio accuracy as show i Figure 22 Mechaics Resoutio ad Treatmet Method o the Midde Suorts The mutisa cotiuous dee beam was divided ito some thi storey beams accordig to the coditios of the disacemet cotiuity ad the stress equiibrium betwee the adjacet iterfaces of the structura ayers The the midde suorts of the dee beam were σ x x + τ xz z =, τ xz x + σ z z = (2) Physica equatio of the rth storey beam: ε x = E x (σ x μ σ z ), ε z = E z (σ z μ σ x ), γ xz = G (τ xz) () (2)

3 Mathematica Probems i Egieerig 3 (3) Geometric equatio of the rth storey beam: ε x = u x, ε z = w z, γ xz = u z + w x, (3) where u ad w are the disacemet of a certai oit aog the directio of x ad z i the beam (4) State equatio estabishmet of the rth storey beam We estabish the first derivative state equatio of about Z directio with (), (2), ad (3): { { { u z σ z z w z τ xz z x G } x = μ x E x μ 2 [ u σ z [ w ], E z E z [ τ xz ] 2 } [ E x x 2 μ ] x } [ ] (4) σ x =E x u x +μ σ z (5) (5) Itroductio of Fourier series Suose u { σ z } { = { w } { τ xz } { { U (z) cos ( πx ) σ (z) si ( πx ) } W (z) si ( πx, (6) ) τ (z) cos ( πx ) } } where U (z), W (z), σ (z), adτ (z) are the comoets of the disacemet ad the stress From (6)ad(5), we ca get the state variabe σ x σ x =E x +μ U (z) si ( πx ) ( π ) σ (z) si ( πx ) It is obvious that (5) ad(6) a meet the boudary coditio, σ x (, z) =, σ x (, z) =, w(, z) =,adw(, z) = (7) (6) Soutio of state equatio of the rth storey beam To ut (6)ito(4), we ca get the foowig: z π G π U (z) { σ (z) } = { W (z) πμ } { τ (z) E x μ 2 } E z E z E [ x 2 π 2 2 πμ ] [ ] [ U (z) σ [ W (z)] [ τ (z) ] It is obvious that the coefficiet matrix i (8) hasbee trasated ito the costat coefficiet matrix; suose (8) π G π K = πμ E x μ 2 (9) E z E z E [ x 2 π 2 2 πμ ] [ ] The (8) is simified ito U (z) U (z) { σ (z) } = {K { W (z) } [ σ (z) [ } W (z)] () { τ (z) } [ τ (z) ] Accordig to matrix exoet method, the soutio of () is obtaied Suose U (z) { σ (z) } =e {K} z [ U () σ { W (z) [ } W ()] () { τ (z) } [ τ () ] R (z) =[U (z),σ (z),w (z),τ (z)] T, D (z) =e {K } z, R () =[U (),σ (),W (),τ ()] T The () issimified ito (2) R (z) =D (z) R () (3)

4 4 Mathematica Probems i Egieerig Whe z=h r,(3) is trasformed ito R (h r ) =D (h r ) R ( r ), (4) where r ( r = ) is the coordiate vaue of the rth storey beam to, h r is the coordiate vaue of the rth storey beam bottom (h r is equa to rth storey s beam tota height), R ( r ) is the coum vector of the disacemet ad the stress comoets at the ocatio of z = r,adr (h r ) is the coum vector of the disacemet ad the stress comoets at the ocatio of z=h r 24 State Equatio Combiatio Wecaikuathe storey beams accordig to the coditios of the disacemet cotiuity ad the stress equiibrium betwee the adjacet iterfaces of the structura ayers by (4), ad whe r = r K,wecaget The frist storey beam :R (h )=D (h ) R ( ), The rth storey beam :R (h r )=D (h r ) R ( r ), The kth storey beam :R (h K )=D (h K ) R ( K ), R (h r )=R ( r+ ) (5) From (5), we iku each storey beam fromto to bottom: The frist ayer :R (h )=D (h ) R ( ), The frist ayer to the rth ayer : R (h r )=D (h r ) D ( 2 ) D ( ) R ( ), The frist ayer to the kth ayer : R (h K ) =D (h K ) D (h K ) D ( 2 ) D ( ) R ( ) (6) The ast row i (6) is the state equatio of the whoe comosite structure Suose, D (h i ) = D (h K ) D (h K ) D ( 2 ) D ( );thethestatemodeofthewhoestructurecabe writte as R (h K )= D (h i ) R ( ) (7) 25 Discussio of Mode Equatio s Sovabiity () Kow Quatity ad Ukow Quatity i State Comoets Equatio (7)caberewritteas D (h i ) U (h K ) 2D (h i ) { σ (h K ) } = { W (h K )} 3D (h i ) { τ (h K ) } [ 4D (h i ) [ 2D (h i ) 22D (h i ) 32D (h i ) 42D (h i ) 3D (h i ) 23D (h i ) 33D (h i ) 43D (h i ) 4D (h i ) 24D (h i ) [ U () σ 34D (h i ) [ W ()] (8) [ τ () ] 44D (h i ) ] ] The coefficiet matrices i (8) are a costat matrices, ad the ukow quatities [U (h K ) σ (h K ) W (h K ) τ (h K )] T ad [U () σ () W () τ ()] are the stress ad disacemet comoet vaue o the to ad bottom of the dee beam (2) Boudary Coditio We take the two-sa cotiuous dee beam; for exame, we estabish the oad boudary coditiosofthetoadbottombeamifigure The boudary coditio of the dee beam s to surface is (x) =σ z (x, ) τ xz (x, ) =, (9) where (x) is the cocetrated oad o the dee beam s to surface The boudary coditio of the dee beam s bottom surface is F (x) =σ z (x, h K ) τ xz (x, h K )=, (2) where F(x) isthesuortforceofeachmiddesuort The defectio of the jth midde suort of the cotiuous beam meets w(x j,h K )=, (2)

5 Mathematica Probems i Egieerig 5 where x j is the x directio coordiate of the jth midde suort, j= m, ad w(x j,h K )= W (h K ) si ( πx j ) (22) The cocetrated force (x) o the to surface of the beam ca be exaded ito the form of Fourier series (x) = { 2 [ u si ( πx i )] si ( πx )}, (23) where x i is the x directio coordiate of the ith cocetrated forceothetosurfaceofthebeam,adi= u (3) Soutio of Ukow Quatity i State Comoet Accordig to the aforemetioed boudary coditios above ad the Fourier series exasio of the mechaica quatities, we ca sove the exressios for the stress comoets [σ (), τ (), σ (h k ), τ (h k )],adthecocretesovigrocess as foows: σ z (x, ) =(x) = { 2 = 2 τ xz (x, ) == [ u σ () si ( πx ) si ( πx i )] si ( πx )} σ () [ u σ z (x, h K )=F(x) = τ xz (x, h K )== si ( πx i )], τ () cos ( πx ) τ () =, σ (h K ) si ( πx ), τ (h K ) cos ( πx ) τ (h K )= (24) The defectio of the jth midde suort w (x j,h) == W (h K ) si ( πx j ) W (h K ) = (25) Oy σ (h K ) is the ukow variabe i the stress comoets [σ (), τ (), σ (h k ), τ (h k )], butσ (h K ) ca be soved accordig to zero defectio o the bottom surface of the midde suort So the stress comoets [σ (), τ (), σ (h k ), τ (h k )] are a kow variabes, ad the other two comoets [U (), W ()] ca aso be soved with (8) So the stress comoets R () = [U (), σ (), W (), τ ()] of the dee beam s to surface areasovedthedisacemetadthestresscomoet vaue of the cotiuous dee beam ca be a soved with (8), ad uttig the soved comoets ito the Fourier series exressios of the disacemet ad the stress, so the mechaica arameters of the mutisa cotiuous dee beam ca be aso soved Defectio (mm) 7 m m = 25 t 2 4 m = 25 t 65 m Figure 4: Schematic diagram of the beam This aer Sa (m) Figure 5: The defectio curve of the bottom of the two-sa cotiuous dee beam 3 Exame 3 Exame Itroductio The beam height is 65 m, the cocrete stregth is C4, the beam egth is 3 m, the comutatioa sa of the sige sa is 4 m, the beam width is 2 m, the stee bar diameters are Φ2 ad Φ8, resectivey, ad the oad P o the dee beam s to is 25 t (Figure 4) 32 Comariso ad Discussio I order to test the accuracy of the theoretica mode, we comared the theoretica mode resuts with () Defectio Curves Figure5 shows the defectio curves of the two-sa cotiuous dee beam The cacuatio resuts of the theoretica mode are cosistet with those of, adthetwocurvesoveraedtogether,whichcarovethe vaidity ad aicabiity of this estabished mode ad aso suggest that the estabished mechaica mode ca be used to aayze the mechaica behavior of the mutisa cotiuous dee beam (2) Shear Stress Distributio Curves aog the Beam Height Figure 6(b) shows the shear stress distributio aog the beam height (as Figure 6(a) shows the ocatio of the observed cross-sectios) It is quite obvious that the theoretica resuts are cosistet with resuts, ad there is o shear stress othetoadbottomsurfaceofthestructure,whichis cosistet with the fact (3) Trasverse Strai Distributio Curves aog the Beam Height Figure 7 shows the trasverse strai distributio aog the beam height (as Figure 6(a) shows the ocatio of the observed cross-sectios) The agreemet is good i Figure 7,

6 6 Mathematica Probems i Egieerig (a) Shear stress (kpa) Shear stress (kpa) Shear stress (kpa) Shear stress (kpa) The 2d sectio The 3rd sectio The 4th sectio The 5th sectio Shear stress (kpa) Shear stress (kpa) Shear stress (kpa) Shear stress (kpa) This aer This aer This aer This aer The 6th sectio The 7th sectio The 8th sectio The 9th sectio (b) Figure 6: (a) The ocatio of the observed cross-sectio (b) The shear stress curves aog the beam height ad the oy error for mode ad the theoretica mode is because of the differece i mesh geeratio, mode simificatio, ad arameter seectio I the reset study, the trasverse strai distributio curves are ot cosistet with the iear reatio but the curve reatio; the same as the other research studies, which shows that the dee beam o oger matches the ae sectio assumtio [7] The reaso is that there are the shear stress ad the orma stress i the

7 Mathematica Probems i Egieerig Strai (με) Strai (με) Strai (με) Strai (με) The 2d sectio The 3rd sectio The 4th sectio The 5th sectio Strai (με) Strai (με) Strai (με) Strai (με) This aer This aer This aer This aer The 6th sectio The 7th sectio The 8th sectio The 9th sectio Figure 7: The trasverse strai curves aog the beam height cross-sectio of the beam, ad the shear stress ca ead to the cross-sectio warig whie the orma stress ca ead to the sectio i arae with the eutra ayer extrusio Figure 8 shows the curve of the trasverse strai distributio aog with the differet beam height (h = 65m,h = 55 m, h =45m,h =35m,h =25m)iSectio2 With the beam height chages, the trasverse strai curve turstobetheiearreatioadmatchestheaesectio assumtio graduay, that is because the beam beogs to thecotiuousshaowbeamisteadofthecotiuousdee beam with the decrease of the beam height ad the ae sectio assumtio ca be used agai The atter i Figure 8 is same to the former research achievemets [7] which roved the vaidity of the cacuatio mode agai (4) Vertica Comressio Deformatio Curves of the Differet Grouds i the Midde Suort Sectio Figure 9 shows the vertica comressio deformatio curves of the differet oits i the midde suort sectio Whe the beam height goes higher, the vertica comressio deformatios of the differet oits are bigger i the midde suort sectio ad whie the beam height is ower, the vertica comressio deformatios are smaer With the icrease of beam height, the vertica comressio deformatios imact o the bedig deformatio of the mutisa cotiuous dee beam icreasigy 4 Cocusios I this aer, we exoited the state sace theory to aayze the mutisa cotiuous dee beam ad ut forward a ew waytosovethestructuredesigofthemutisacotiuous dee beam, which wi earge the aicatio scoe of the state sace theory The mode was estabished without the ae sectio assumtio ad the stress ad the disacemet assumtio, so it is more efficiet i the estabishmet ad cacuatio tha mode; aso it has wider aicatio rage ad higher recisio, eseciay i the cacuatio of the arge comosite structure This aer reveas that there are the shear stress ad the orma stress i the cross-sectio of the beam; the shear stress ca ead to the cross-sectio warig ad the orma stress ca ead to the sectio i arae with the eutra

8 8 Mathematica Probems i Egieerig h = 65 m h = 55 m h = 45 m 3 2 h = 35 m h = 25 m Strai (με) Strai (με) Strai (με) Strai (με) Strai (με) Figure 8: The trasverse strai curves aog the beam height of differet beam The oits i the sectio of the midde suort Defectio (mm) h = 65 m h = 55 m h = 45 m h = 35 m h = 25 m Figure 9: The vertica comressio deformatio curves of the differet grouds i the midde suort sectio ayer extrusio With the icrease of beam height, the vertica comressio deformatios ca imact o the bedig deformatio of the mutisa cotiuous dee beam icreasigy The cacuatio soutio is cosistet with soutio, ad the mechaics atter reveaed by the mode is simiar to the former research achievemets which suggests that the mode ca be used for ractica cacuatio Refereces [] Z-Z Wag, J-Z Zhu, L Che, J-L Guo, D-Y Ta, ad W-J Mi, Stress cacuatio method for dee beams with shear-bedig couig distortio uder cocetrated oad, Egieerig Mechaics,vo25,o4,5 2,28 [2] G R Cower, The shear coefficiet i timosheko s beam theory, Joura of Aied Mechaics,vo33,o2,335 34, 966 [3] SPTimoshekoadJMGere,Stregth of Materias,VaostradComay,NewYork,NY,USA,972 [4] R D Midi, Ifuece of Rotary iertia ad shear o fexura motios of isotroic, eastic Pates, Joura of Aied Mechaics,vo8,3 38,95 [5] M Leviso, A ew rectaguar beam theory, Joura of Soud ad Vibratio,vo74,o,8 87,98 [6] M Leviso, Further resuts of a ew beam theory, Joura of Soud ad Vibratio,vo77,o3,44 444,98 [7] Y Gao ad M Z Wag, Refied theory of symmetry deformatio rectaguar beam, Sciece i Chia G, vo4,57 522, 29 (Chiese) [8] BYYag,XTWu,adHPLi, Researchotherecisioof the umerica cacuatio of the disacemet of short or dee beams with shear ad bedig, Joura of Aied Mechaics, vo2,o2,45 46,23 [9]PLMeiadDSZeg, Precisesoutioofdeebeams, Mechaics ad Practice,vo24,o3,58 6,22(Chiese) [] G F Wag, Stress aaysis of simy suorted dee beams, Joura of Chegdu Uiversity of Sciece ad Techoogy,vo7, o3,7 76,993(Chiese)

9 Mathematica Probems i Egieerig 9 [] D J Dig ad W Q Liu, Soutio of dee beams based o mechaics of bar system, Egieerig Mechaics,vo,o, 8, 993 (Chiese) [2] L Z Che ad Z Z Li, Series soutio of cotiuous dee beam uder distributed oad, Desig ad Research,vo32,o 9, 3 32, 25 (Chiese) [3] Z Z Wag ad J D Sha, Cacuatio of orma stress ad defectio i trasverse bedig of mai beam i outet gate, Joura of Hydrauic Egieerig, vo 9, 4 46, 995 (Chiese) [4] CMWag,KYLam,XQHe,adSChucheesaku, Large defectios of a ed suorted beam subjected to a oit oad, Iteratioa Joura of No-Liear Mechaics,vo32,o, 63 72, 997 [5] Y-C Jiag, X-F Hu, ad H Che, The series soutio of sum fuctio method for sime suorted dee beam, Joura of Sichua Uiversity,vo38,o6,63 67,26(Chiese) [6] Z Z Wag ad L C Li, Critica sa-deth ratio of thiwaed stee beam, Mechaics i Egieerig, vo9,o4, 23 24, 997 [7] X Y Wag, Exerimeta study ad fiite eemet aaysis o ew stee reiforced cocrete beam [PhD dissertatio], Cetra South Uiversity, 27

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