THEORY OF THE FOUR POINT DYNAMIC BENDING TEST PART IV: PURE BENDING & SHEAR DEFORMATION

Transcription

1 THEORY OF THE FOUR POINT DYNAMIC BENDING TEST PART IV: PURE BENDING & SHEAR DEFORMATION IMPORTANT NOTICE March 8 & 9 ; 7 I the origial report the coefficiet i the formula for the effective cross area (deoted by: { BH, }, for the shear force), is take equal to /3. Just before the 1 st Europea PB workshop o March 8 & 9; 7 it was cocluded, based o ABACUS FEM calculatios, that a value of,85 was more appropriated ad leads to: { BH, } =,85.B.H

2 THEORY OF THE FOUR POINT DYNAMIC BENDING TEST PART IV: PURE BENDING & SHEAR DEFORMATION AUTHOR: A.C. PRONK DATE: 1 ST CONCEPT SEPTEMBER FINISHED MAY 7

3 Abstract This report deals with the commoly eglected iflueces of shear forces o the measured deflectios i the dyamic four poit bedig test. The report forms the fourth part i the series of reports o the Theory of the Four Poit Dyamic Bedig Test. It is world wide adopted that for a bedig test i which the ratio of the (effective) legth or spa of the beam ad the height of the beam is above a factor 8 the deflectio due to shear forces ca be eglected. This judgemet is based o a compariso betwee the differetial equatios for pseudo-static bedig tests. I this report the complete aalytical solutios are derived for cyclic bedig test coditios. It is show that the deflectio part due to shear forces is aroud 5% of the al deflectio. For a good uderstadig of the theory it is advised to use Part I ad II of the series o the Theory of the Four Poit Bedig Test et to this report. Cotets 1. GENERA BENDING THEORY OF A RECTANGUAR BEAM. BOUNDARY CONDITIONS 3. DEVEOPMENT OF Q{,t} IN SERIES OF SINES (orthogoal fuctios). DEVEOPMENT OF THE OTHER MASS FORCES IN SERIES 5. FUNCTION FORMUATION FOR Vb ad Vs 6. THEORETICA SOUTION WHEN NO EXTRA MASSES ARE PRESENT 7. SOUTION FOR THE PSEUDO-STATIC CASE 8. HOW TO INCORPERATE EXTRA MOVING MASSES. ANNEX I Developmet i Sie or Cosie Series ANNEX II The series developmet for the deflectio due to shear DISCAIMER This workig paper is issued to give those iterested a opportuity to acquait themselves with progress i this particular field of research. It must be stressed that the opiios epressed i this workig paper do ot ecessarily reflect the official poit of view or the policy of the director-geeral of the Rijkswaterstaat. The iformatio give i this workig paper should therefore be treated with cautio i case the coclusios are revised i the course of further research or i some other way. The Kigdom of the Netherlads takes o resposibility for ay losses icurred as a result of usig the iformatio cotaied i this workig paper.

4 Pure Bedig & Shear Deformatio 1. GENERA BENDING THEORY OF A RECTANGUAR BEAM The bedig theory for a rectagular beam is give by two differetial equatios:.b.h. Vb,t Vs,t D Q,t t 3.I. Vb,t M D t [1a,b] I which: Vb = Deflectio due to pure bedig Vs = Deflectio due to shear forces B = Width of beam H = Height of beam = Total legth of beam = Effective legth = distace betwee the two outer supports A = Distace betwee outer ad ier supports = Distace from = to first outer support (=(t-)/) F o = Applied force at the two ier supports (clamps) I = Momet of the beam (B.H 3 /1) D = Shear force M = Bedig momet Q = Force distributio alog the beam E = Stiffess modulus of beam G = Shear modulus (E/((1+))) = Desity = Poisso ratio The momet M is related to the deflectio Vb by: M E.I. Vb,t [] Ad the shear force D is related to the shear deflectio Vs by: D G. B,H. Vs, t with B,H Effective Cross Area [3] Itegratio of equatio [1b] over leads to a relatioship betwee the shear deflectio Vs ad the bedig deflectio Vb: Vs I,t.. Vb,t E. Vb,t t [] G. B,H t i which t is a yet ukow fuctio i the time t oly.

5 Differetiatio of equatio [1 b ] ad addig to equatio [1 a ] leads to a relatioship betwee the al deflectio Vt ad the bedig deflectio Vb :. B.H. Vt,t.I. Vb,t E.I. Vb,t Q,t [5] t t Usig equatio [] i equatio [5] leads to the desired relatioship betwee the bedig deflectio Vb ad the force distributio Q:.B.H. t.b.h.i G. B,H Vb,t.I Vb,t E.I. Vb,t.. t Vb t t d dt,t E. Vb,t t Q,t [6] Elimiatio of Vb by substitutig equatio 1a ito equatio 1b after two derivatios with respect to the time t leads to: E.I.B.H d. d Q t,t.b.h. Vs,t G. B,H. Vs,t.I d.b.h dt Q t,t.b.h. Vs,t G. B,H Vs,t t [7] G t d dt B,H. Vs,t t From equatios [6] ad [7] it is clear that both Vb ad Vs cotai a fuctio which oly depeds o the time: Vb+f{t} ad Vs+g{t}. Rearragemet of the two basic differetial equatios leads to: E.I. Vb{,t }.B.H. Vb{,t }.I. Vb{,t } t t [8].B.H..I. G. { B,H } t Vb{,t } E.I. t Vb{,t } Q{,t } I. t G. { B,H }..B.H G. { B,H }. t. t Vs{,t } I.B.H E... t. Vs{,t } E..Q{,t } [9]

6 After estimatios for the several coefficiets the followig two (approimated) differetial equatios for the pure bedig deflectio Vb ad the shear deflectio Vs ca be established (see referece 1): E.I. Vb{,t }.B.H. Vb{,t } t G. { B,H }. Vs{,t } Q{,t } Q{,t } [1a,b] Equatios 1 a ad 1 b are used for the forward calculatio of the deflectio V = Vb+Vs if the other parameters like E, B, H, G, ad Q{,t} are give. Notice that Q{,t} has the dimesio N/m; i equatio 1 it is a force per legth. The deviatio made by the igorig the other terms i equatios 8 ad 9 is egligible. Mark that if etra movig es are preset, the two differetial equatios are coupled by a etra term (commo force) i Q{, t} F{, t}. { } M. Vt {, t}. { } t both equatios: [1 c ] M. Vb {, t} Vs{, t}. { } t This epressio has the dimesio N/m ad acts oly at = ad ad at = ad. See also equatio [1] ad [13]. Equatios 1 a ad 1 b will be solved (the steady state) for a siusoidal load sigal of the form: e it.. BOUNDARY CONDITIONS I case of a four poit bedig test (PB) the followig boudary coditios are valid for equal siusoidal poit loads at the two ier clamps ad oly regardig the steady state coditio: At ad : M{,t} E.I. Vb{, t} & D{,t} G. {B,H}. Vs{, t } [I] At ad - : Vt{,t} = Vb{,t} + Vs{,t} [II] At / : Vb{, t } & Vs{, t } [III] [11a,b,c]

7 3. DEVEOPMENT OF Q{,t} IN SERIES OF SINES (orthogoal fuctios) For the solutio of the differetial equatios it is ecessarily 1 to develop the discrete positio fuctios for the poit loads ito series of orthogoal fuctios. Give the simple character of the equatio a commo sius or cosies series is already sufficiet. The discrete positio fuctio for the load distributio Q{,t} is give by: F.ω. t i Q{,t }. δ Δ δ Δ A δ Δ A δ Δ. e [1] i which {=a} represets the delta fuctio: a. f.d f a a a [13] I case of oly pure bedig a series of orthogoal sie fuctios would be sufficiet. At = ad = the momet M (:: / ) has to be zero, which i case of a sie fuctio is it it automatically fulfilled: Q{, t} F. e F. Si... e [1] 1,3,5,. Because of symmetry oly odd umbers for are allowed. If istead of bedig oly shear deformatio occurs, a developmet i cosie fuctios is more appropriate because ow the shear force D (:: / ) has to vaish at = ad =. it Q{, t} G G. Cos... e. [15],,6,.. I view of the symmetry ow oly the eve umbers for are allowed. Determiatio of the coefficiets F ad G is based o the orthogoal property of the siusoidal fuctios o the iterval from to, which is represeted by the followig equatios: Si m.π. Cos m.π..si.π..cos.π..d for m for m.d [16] It should be oted that because of symmetry oly odd umbers are allowed i case of the developmet i sie fuctios ad oly eve umbers i case of the developmet i cosie fuctios. A combied form (e.g. a Fourier series) is ot possible because the orthogoal property (a costat value) of the series does t hold for the product of a sie fuctio ad a cosie fuctio. Istead of a costat value (equatio 16) the itegratio depeds o ad m: 1 Whe o etra movig es are preset the solutio of the differetial equatio 1b which describes the deflectio Vs due to shear ca be obtaied directly as the product of the time fuctio ad the solutio for the (pseudo) static case. This last solutio ca be easily foud by itegratio over the several itervals. This is due to the fact that without a couplig by etra movig es the phase lag i the deflectio Vs has to be equal to the (opposite) phase lag of the comple stiffess modulus Smi of the beam.see also Ae I & II.

8 Si m.π..cos.π..d π for m. m m eve for m odd [17] The discrete poit load fuctios ca ow be trasformed i either a series of sie fuctios (pure bedig) or i a series of cosie fuctios (shear deformatio): F F. Si.. F. F Si d.... i t Q t 1 {, }.e with G G. Cos.. G. G. Cos... d [18] Furthermore it should be marked that the value = is ot possible for the developmet i cosie fuctios (see the equatio for the shear deformatio; dividig by ). Replacig F{ } ad G{ } i equatio [18] by the epressio for Q{, t} with the delta fuctio { }leads to equatios [19] ad [] for the coefficiets F ad G. The equatio for the coefficiet F is also give i part II of this series (usig the symbol A ). A Si.. Si.. F F. A Si.. Si.. A Cos.. Cos.. F G. A Cos.. Cos.. F A F.. Si.. Si.. for 1,3,5,7 etc. F A G.. Cos.. Cos.. for,,6 etc [19] [] I accordace with the formulatios used i Part II of this series the followig abbreviatios are itroduced: A T { } Si. Si Si [1] A U{ } Cos. Cos Cos []

9 If = eff (=) tha i accordace with earlier otatios i part I of these series the followig fuctios will otatios will be used: T {} P {} ad U {} R {}. Mark that for = the coefficiet G ad the fuctio U {} are give by (=,,6,..):. F A G.. Cos.. 1 A U { } R { } Cos Cos. 1 ; [3] I this way the poit loads at the ier ad outer supports ca be take ito accout by trasformig those four poit loads i a force distributio Q{,t} alog the beam. The force distributio Q{,t} ca be either represeted as F iωt iωt Q{,t}=. T { }.e ( odd ) or as Q{,t}=. U { }.e ( eve ) 1 F []

10 . DEVEOPMENT OF THE (OTHER) MASS FORCES IN SERIES First of all it should be metioed that the placig of a sigle etra at a arbitrarily chose locatio (ecept = /) will ot be dealt with. Oly symmetrically placed es will be take ito accout. Normally these etra es are located at the ier clamps ( of the pluger etc.). These etra forces will like the eteral drive force raise reactio forces at the outer clamps. If the forces are deoted by F / at ad F / at -, the force distributio Q { mas,,t} will be give by: Q{, t}. F. Si... d. Si... e 1,3 F 1,3 it Si... Si.. Si... e it [5] for implemetatio i the pure bedig equatio (1 a ) ad for the shear equatio(1 b ) by: Q{, t}. F. Cos... d. Cos... e, F, it Cos... Cos.. Cos... e it [6] Notice the differece with the epressio for the (drive) force distributio Q{,t}, if is ot equal to the locatio of the ier clamp: A+. To get i lie with the epressio for this force distributio Q{,t} the followig coefficiets are itroduced: Si Si T { } A Si Si F Q{, t} Si... Si.. Si.. 1,3 F T { }. T{ } 1,3 Cos Cos U{ } A Cos Cos F Q{, t} Cos... Cos.. Cos.., F U { }. U{ }, [7a,b]

11 Epressio for the forces due to movig es at the ier clamps Q t M Vt A e M Vt A e t. T { } it iclamp it i. clamp clamp{, } clamp. { }. clamp { }.. it i. clamp MclampVt{ A}.. e. or 1. U { } If equals A+ tha of course T { } = U { } = 1. [8] Epressio for the force due to a movig at the cetre Q t M Vt e M Vt e t. T { }. T { } it iceter it i. ceter ceter{, } ceter. { ceter }. ceter { ceter }.. ceter 1 it i. ceter MceterVt{ ceter }.. e. or. { }. { } U ceter U [9]

12 5. FUNCTION FORMUATION FOR Vb ad Vs Whe etra movig es are icluded the origial differetial equatios 1 a,b have to be rewritte as: E. I. Vb{, t}. B. H. Vb{, t} Q{, t} Q{, t} t G. { B, H}. Vs{, t} Q{, t} Q{, t} [3a,b] The load distributios alog the beam are give by: F iωt iωt Q{,t}=. T { }.e ( odd ) or as Q{,t}=. U { }.e ( eve ) 1 F [31] F F Q {, t} T { }. T { } or as Q{, t} U{ }. U{ } [3] 1,3, The parameter F is give by: d d F M Vt{, t} M Vb{, t} Vs{, t} MVt{, t} d t d t [33] All forces are writte i the form of sie or cosie series usig T {} or U {}. Therefore it is logical to assume the followig epressios for Vb ad Vs: Vb{,t } Vs{,t } iω t iφ iω t Va{ } Vc{ } Vd { }.e A e T. { } Vc{ } Vd { }.e 1 1 iω t iφ iω t Vg{ } H.e H e U. { } H.e [3] [33] The fuctio Va{} satisfies the complete differetial equatio icludig the forces iduced by etra movig es. The fuctios Vc{} ad Vd{} are solutios of the homogeous differetial equatios without the eteral force ad iduced forces. They are eeded to satisfy the boudary requiremets at the outer supports as will be show later o. The fuctio Vg{} plays the same role as Va{} i the differetial equatio for the shear deformatio. The costat H ( beig a costat because at = the shear force must be zero) is comparable i fuctio to Vc{} ad Vd{} ad is also eeded because at the outer clamp the al deflectio Vt{,t}=Vb{,t}+ Vs{,t} has to vaish. The deflectio Vb{,t} equals zero because Va{} + Vc{} + Vd{} =.

13 The fuctios Vc{} ad Vd{} are coupled to each other because they have to meet (together) the requiremet for the bedig momet of Vb{} at = ad =. Also Vs{,t} has to be zero, which is accomplished by Vg{} + H =. Remember that Vb{,t} = Vb{ -,t} ad Vs{,t} = Vs{ -,t}. 6. THEORETICA SOUTION WHEN NO EXTRA MASSES ARE PRESENT I this case the two differetial equatios are ot coupled by a etra movig at a arbitrarily locatio. For a viscous-elastic material the equatios ca be writte as: F S e I Vb t B H Vb t Q t T e mi i it {, } {, } {, } 1{ }. t 1 Smi F e B H Vs{, t} Q{, t} U { }. e ; Note : is take Real [3] (1 ) i it 1 Vb{, t} Vb{ }. e ; Vs{, t} Vs{ }. e Bedig Deflectio Vb it it The solutio for Vb is ow represeted by (.B.H. = M beam ): Vb{ } Va{ } Vc{ } Vd{ } Va, 1{ } Vc{ } Vd{ } 1 1 i A e 1 1 T 1{ } Vc{ } Vd{ } With A F i1 1e i 1 mi. 3 S e I M beam The similarity with a -sprig system is already obvious. ater o the epressios will be give which are used i the Ecel program. For the complete solutio the fuctio Vc{} ad Vd{} are eeded whe the al legth of the beam is bigger tha the distace betwee the two outer clamps: [35] i Vc{ } C1 Cos e 1 [36] The fuctio Vc{} satisfies the homogeous differetial equatio (without the force distributio Q{}). Because the differetial equatio is of the order, the complemetary fuctio Vc{i} is also a solutio (i =+1). i i Mbeam Vd{ } D Cos i e D Cosh e with 1. 1 ; 1 1 Smi I [37]

14 Fially the followig requiremets have to be fulfilled: Requiremets : Va{ }+Vc{ }+Vd{ } = ad Va{ } Vc{ } Vd{ } [38] The secod requiremet leads to the followig relatioship betwee C -1 ad D -1 : Cos Va{ } D 1. C1 C1 Cosh [39] Ivokig i the first requiremet leads to: i 1 A 1. e T 1{ } C1 Cos D 1 Cosh i 1 or A 1. e T 1{ } C1 Cos Cosh Shear Deflectio Vs [] Importat: As metioed before, due to the lack of etra movig es the solutio ca be obtaied as the product of the time fuctio e it ad the static solutio. The last oe ca be established by a simple (double) itegratio of the differetial equatio i which the poit loads are represeted as the product of the load ad a Dirac fuctio (). However, whe a couplig eists betwee the two differetial equatios this solutio procedure will fail. I that case a developmet i sies or cosies has to be used as doe i this chapter. See also Ae I ad II. Give the fact that the first derivate has to be zero at = ad at = the deflectio Vs is chose as : i g, 1 1 Vs{ } V { } H e U { } H F (1 ) i He.. ; Notice that i Smi e B H ( ) [1] i Requiremet Vs{ } = H H e U 1 { } I this case the load distributio is developed ito a cosie series. Therefore the requiremet that the shear force at = must be zero is automatically fulfilled. By addig a costat the requiremet that the deflectio has to be zero for = ca be met.

15 It is also possible to obtai a alterative solutio epressio for Vs 3 : Alterative : Vs{ } V { } H e T { } H H ; i 1 g, F (1 ) i 1 H1 e.. ; Notice that i 1 Smi e B H ( 1) H 1 Vg, 1{ } 1 t ot ; { } i 1 H1 H H 1 e T 1 1 F (1 ) ( 1) H H H.... T { / } 1 1,1 1,1 i 1 1 Smi e B H ( 1),1,1 i 1 mi H H H F (1 ).. T T 1 1 S e B H. { ( 1) } {} [] The alterative solutio is epressed i the odd T -1 {} series used i the solutio for pure bedig ad could be from this poit a attractive alterative. However, this solutio require much more terms compared to the basic solutio usig the eve U {} series. Calculatio of Shear Deflectio U Series: Cos() T Series: Si() 6,E-6 5,E-6,E-6 Shear Deflectio Vs 3,E-6,E-6 1,E-6,E+ -1,E ,E-6-3,E-6 Figure 1 Number of terms The developmet of the Shear Deflectio usig a Cosie (U) ad a Sie (T) series 3 Whe the load distributio is developed ito a sie series, two solutios for the homogeous differetial equatio are eeded i order to fulfil both the requiremets at = ad =. Furthermore the derivate of T{} at = equals {./}.T{/} If = the T-1 series aloe do ot satisfy the requiremet for the shear force at =. The T-1 series will lead after derivatio to a series of 1/(-1) terms. I order to satisfy the boudary restrictio a solutio of the homogeous differetial equatio has to be added: - /(-1). This solutio will elimiate the shear force after the derivatio of the terms /(-1) at =. However, the series 1/(-1) coverges badly.

16 It is also possible to epress Vs i Vb usig equatio 1 a or equatio. However equatio cotais a ukow fuctio i t ad equatio 1 a is i fact similar to equatio 1 b because the first term i equatio 1 a ca be eglected compared to the secod term. Therefore i the Ecel program oly the fial solutio for Vt is obtaied by developig Vs i cosie series. Total egth equals the Effective egth eff (=) I that case the basic series developmet T ad U are defied by the symbol P ad R : If = ( =) tha: eff A T 1{ } P 1{ } Si 1. Si 1 ; P 1{} eff eff A A U{ } R { } Cos. Cos 1 ; R{} Cos 1 eff eff eff Vb{ } i1 Va, 1{ } A 1 e P 1{ } ; Vc{ } Vd{ } 1 1 i i A Vs{ } Vg,{ } H e R{ } H ; H H e Cos eff [3] 7. SOUTION FOR THE PSEUDO-STATIC CASE I the pseudo-static case the iertia forces do ot play a role. Therefore the ratio of Vs ad Vb ca easily be determied from equatio [] by omittig the time depedet terms. d I E. Vb Vs d I which E is the Stiffess of the beam [] G. B, H For (A ) the bedig deflectio Vb is : 3 F ( ) A A Vb = E I d I E. Vb F A F A d A(1 ) 3 Vs ; Vb G. B, H E. B, H 1 E B H 3 Vs F H. A 1. Vs{( A ) }. G { B, H }. 3 A Vb [5] 5 [6] Vs Vb 5 I ASTM stadards A=/3; Usig = /3 the ratio will be:. 1 μ 5 3. H

17 For A the bedig deflectio Vb is : 3 F ( ) ( ) A A Vb = E I d I E. Vb d F (1 ) Vs{ A }.( ) G. B, H E B H [8] For the bedig deflectio Vb is : F ( A) A Vs{ } [9] Vb = EI I may tetbooks ad papers the value for is take equal to /3. However, based o 1D, D ad 3D fiite elemet calculatios with ABAQUS it was foud that a value of,85 was more appropriate. That s to say the aswers from 3D (ad 1D) calculatios are comparable to the aalytical solutios if a value of,85 was used. As epected the ratio depeds o 3 parameters: Poisso ratio, the ratio H/ ad the ratio A/. The value is obtaied by comparig FEM calculatios for the 1D case (bar elemets) ad for the D case. I the 1D case it is possible to vary the stiffess modulus E ad the shear modulus G idepedetly. I this way the deflectios due to pure bedig ad shear ca be determied separately (by choosig a ifiite value for G). Assumig that the D case (ad 3D case) will give correct aswers the ratio Vs{ /}/Vb{ /} is determied. For H/ ratio of 5/5; 5/5 ad 1/5 a value of,85 was established. 8. HOW TO INCORPERATE EXTRA MOVING MASSES. The etra iertia forces are preset i both differetial equatios: [7] i Smi e I Vb{, t} B H Vb{, t} M Vt{, t} Q{, t} t t Smi i e B H Vs{, t} M Vt{, t} Q{, t} (1 ) t it it Vb t Vb e Vs t Vs e Vt t Vb t Vs t {, } { }. ; {, } { }. ; {, } {, } {, } [5a,b] I this report a correct solutio procedure will be followed i cotrast with a procedure outlied i Part II Overhagig Beam Eds ad Etra Movig Masses of these series. I Part II the deflectio due to shear was igored ad Vt{,t} was take equal to Vb{,t}. By multiple iteratio (calculatig Vb{,t} ad Vb{,t} the ultimate deflectio values were established. This procedure was ad is attractive because for the back calculatio procedure the etra movig es could be icorporate i a easy way. However, if the deflectio due to shear is ot igored a differet forward calculatio has to be performed. First of all the equatios 5 a,b are rewritte as:

18 i Smi e I Vb{, t} B H Vb{, t} Q{, t} M Vt{, t}. { } t t Smi i e B H Vs{, t} Q{, t} M Vt{, t}. { } (1 ) t [51a,b] I this way the forces due to a etra movig at a prescribed locatio are icluded i the eteral (drive) force. I fact the same type of formulas as preseted i chapter 6 ca be used if the force F is replaced by: F F M Vt{ } accordig to equatios [31] to [33]. Because the iflueces of etra es are for ormal coditios small, a simple iteratio procedure ca be performed: 1. The first step is the calculatio of the bedig ad shear deflectios at the desired locatio ad the locatio(s) where the etra movig es are usig the equatios for the system without etra movig es.. Tha the al deflectio at is calculated: Vt{ } = Vb{ } + Vs{ } 3. The secod step is the calculatio of the deflectios with etra movig es i which for the deflectio Vt{ } at the right had side of equatios [51 a,b ] the value of the former step is take.. Step 3 is repeated util the estimate value for Vt{ } of the former step does t differ from the ew estimated value for Vt{ } 5. I the Ecel program the iteratios are ot performed usig e.g. Visual Basic. Therefore the umber of iteratios is limited to ie steps which are i ormally situatios eough.

19 ANNEX I Developmet i Sie or Cosie Series I chapter 5 the load fuctio Q was developed ito ifiite a series of sie or cosie fuctios. With respect to the boudary coditios, the sie fuctio series was adopted for the deflectio due to pure bedig ad the cosie fuctio series was adopted for the deflectio due to shear. I this ae we will try the opposite. Pure Bedig Whe a cosie fuctio series is used the boudary coditio for the secod derivatio is ot met. Therefore eve i the case of o overhagig beam eds the solutios Vc{} ad Vd{} have to be added i it Vb{, t} Ae U Vc Vd e. { } { } { }. I-1,,6,.. The other coditio is that at the outer clamp the deflectio should be zero. it i it Vb{, t} Va{ } Vc{ } Vd{ }. e A e. U { } Vc{ } Vd{ }. e I- 1 Therefore we have two equatios, which make it possible to calculate the coefficiets C ad D of the deflectios fuctios Vc{} ad Vd{}. Pure Shear I case of the deflectio due to shear it is also possible to solve the problem usig sie fuctios. However, the oly acceptable solutio of the homogeous differetial equatio for the whole iterval ( ) is a sigle costat H times the time fuctio. This is ot eough to satisfy both coditios: o shear at = ad o deflectio at =. However, a acceptable solutio will be possible if the iterval is divided ito separate itervals: / ad /. I the first iterval we have the etra solutio H 1. + H ad i the secod iterval the solutio - H 1. + H + H 1.. A T{ } Si m 1. Si m 1 Si m 1 m 1 A T { } Cos m 1. Si m 1 Si m 1 I-3 m 1 A At : T{ }. Si m 1 Si m 1 m 1 A At : T { }. Si m 1 Si m 1 I-

20 m m 1 A. Si m 1 Si m 1 m1 m A. Si m 1. Si m 1 Si m 1 I-5 m1 m. Si m 1 A. Si m 1 Si m 1 m1 At = / the deflectio is cotiuous but the first derivate chages from sig. ANNEX II The series developmet for the deflectio due to shear INTRODUCTION I the Hadbook of Mathematical Fuctios (M. Abramowitz & I.A. Segu; Dover publicatios, Ic., New York) a overview is give of summable series cosistig of either sies or cosies. Cos( ) e Si for Cos( ) for 6 3 Cos( ) for Si( ) 1 for 3 Si( ) for Si( ) for II-1 II- The series Si( ) e 1 t Si dt { Clause s Itegral } will coverge for but is ot summable like the sie ad cosie series give above. The Clause s itegral ca oly be epressed i aother series represetatio: 1 1 Si( ) e 1 B for 1 1! ( 1) 6 II-3 6 The coefficiet B is ot give i the Hadbook of Mathematical Fuctios but ca (probably) foud i Tabulatios of the fuctio Si( ) ( ) by A. Ashour ad A. Sabri (Math. Tables Aids Comp., )

21 This does ot eplai the large differece i the covergece of the two solutios (sie or cosie series) for the deflectio due to shear. The series epasio for the deflectios due to shear ad pure bedig cosist out of a product of two sies or two cosies. Such a product ca be reduced to a summatio of subtractio of two cosie terms: Cos( ) Cos( ) Cos( ) Cos( ) Si( ). Si ) & Cos( ). Cos( ) II- The obtaied cosie epasios ca be summed up due to the term i the deomiator of the cosie terms (eq. II-1). I the et paragraph it will be show that both series epasios (sie & cosie) will lead to the same aswer. Notice that the same coclusio yields for the deflectio due to pure bedig i which the deomiator of the terms cotais a term (eq. II-1). Further it is oticed the series epasio for the strai due to pure bedig ( d derivatio of the deflectio Vb with respect to ) is build up out of cosies terms divided by (like the epasio for the shear deflectio) ad therefore coverge fast. This is i cotrast with the (bedig) strai derived i the same way for the shear deflectio. The epasio leads i that procedure to a summatio of cosie terms without a deomiator which depeds o. Shear Deflectio ad Shear Strai usig the Cosie ad Sie series epasio Deflectio If o etra movig es are ivolved the deflectio due to shear ca be easily obtaied from equatio [ ]for the static case which leads for the iterval A+ < < / to: 3 F ( ) A A Vb{} = E I d I E. Vb F A 3 A Vs{ } d ; G. B, H E. B, H 1 E B H F A(1 ) Vb 3 II-5 This meas that the shear deflectio Vs has a costat value o this iterval which also ought to be the summatio of the epasio i cosies or sies terms. A proof will be give for the epasio i cosies terms while this epasio coverge the best. Shear Strai I case of the deflectio due to pure bedig the related bedig (horizotal) strai is derived from a d derivatio of the deflectio with respect to the distace X. I fact this d derivatio will give the momet M (see Part I of these series). I the static case the deflectio polyomial betwee X = A + ad X = - A - is of the order two (parabolic). Therefore the (bedig) strai i this regio will be a costat. The CEN formulas are based o this relatioship by assumig the same parabolic equatio for the amplitude of the deflectio i case of a dyamic situatio. The iertia forces due to the weight of the beam ad other etra (movig) es will disturb this simple relatioship. Assumig that for sigle sie or cosie terms i the series epasios for the deflectio due

22 to pure bedig the d derivatio of the deflectio will lead to the correct strai value for this deflectio compoet, the bedig strais are calculated i the Ecel program. As show by calculatios the differeces i strai values with the static case are rather small for low frequecies ad low es for the beam. Ecept for the case i which the chose locatio is equal or ear to the positio of the clamps (supports). As ca be obtaied from the differetial equatio for the deflectio Vs due to shear, a (theoretical calculated) d derivatio with respect to will lead to a zero value as epected, eve i the case of dyamic loadig (without etra movig es). A shear force does ot lead to a horizotal strai i the beam but will oly deform the cross sectio of the beam. This deductio is based o the formulas for the static ad dyamic case i which o etra movig es are preset. Whe a etra movig is preset (e.g. the weight of the pluger) the two priciple differetial equatios are coupled by the iertia force at the poit where the is coected to the beam (ofte the two ier clamps or supports) ad the reactio forces at the two outer supports. Due to the etra term the phase lag for the shear deflectio will ot have the same (but opposite) value as the phase lag of the comple stiffess modulus for the beam. The differece is small but eists. Therefore the shear strai should be calculated from the deflectios which are based o a epasio i sies or cosies terms. However, especially for locatios at ad ear the clamps will lead to lead to urealistic strai values. Performig the required d derivatio will elimiate the covergece of the epasio (the factor will vaish i the deomiator of the terms). Oly the appearace of a etra movig may give some covergece. Shear Deflectio Vs Omittig the term for the phase lag ad the time fuctio, the solutio i a series developmet is give by equatio II-6 F Vs { } H U { } U { } (1 ) ; H. S B H. II-6 ( ) 1 mi The series U is give by equatio II-7. A U { } Cos Cos Cos. II-7 Usig equatio II- equatio II-7 ca be rewritte as equatio II-8 A U { } Cos Cos Cos. A A Cos Cos Cos Cos II-8

23 A U { } Cos Cos Cos. A A Cos Cos 1 Cos II-9 If equals equatios II-8 ad II-9 ca be rewritte as: A A Cos Cos Cos A { } ; {} 1 U U Cos II-1 Due to the term i the deomiator (see eq. II-6) these series ca be summed up. Applyig equatio II-1 leads for the iterval A+ / fially to the solutio as give by equatio 5: (1 ) F A Vs{ A } II-11 S B H mi As metioed before usig the U {} series will lead to a reasoable fast covergece for the developmet of the series. This is due to the fact that the U {} series already satisfy oe boudary restrictio of the differetial equatio (shear force = for = ). Usig the T -1 {} series is a differet story. Specially whe the covergece is goe. Horizotal Shear Strai Cotributio(?) The title of this paragraph is somehow misleadig. As ca be see by the equatios for the shear deflectio i chapter 7, the shear force will ot cotribute to the horizotal strai i the beam i the case that there are o etra movig es. But eve whe etra movig es are preset, there ought to be o cotributio. However, i that case the strai is calculated by takig the secod derivate of the deflectio. Ad hece the covergece of the series epasio is lost while the term (-1) i the deomiator of the terms is elimiated by the derivatio. Therefore it is better to calculate the strai i aother way: 1. Defie, i spite of the results of the d derivate of the shear deflectio, the cotributio of the shear force to the horizotal strai as zero.. Multiply the calculated or measured al deflectio accordig to the followig formula: Vb { } Vt{ }. Ratio { } 1 1

24 3. For measurig the al deflectio i the cetre (/) the Ratio is equal to: Vs{ } H Ratio 1. { } [Pseudo-static case] Vb{ }. 3 A Ad if the deflectio is calculated or measured at the ier clamp by: Vs{ A} 1. H Ratio{ A } [Pseudo-static case] Vb{ A}. 3 A. I this way a good estimate for the deflectio due to bedig is obtaied. By multiplyig this deflectio with the followig factor a good estimate is foud for the real horizotal strai value. 5. R { } A 3 3 A HA 3H { }. R { }. Vb { }. Vb { } A