Ameran Journa of Cv Engneerng and Arhteture,, Vo., No. 4, 8-9 Avaabe onne at http://pubs.sepub.om/ajea//4/ Sene and Eduaton Pubshng DOI:.69/ajea--4- A Mxed Fore-Dspaement Method for the Exat Souton of Pane Frames Govann Fasone,*, Daro Settner Department of Cv Engneerng, Unversty of Messna, Messna, Itay *Correspondng author: gfasone@unme.t Reeved Apr 4, ; Revsed June 9, ; Aepted June, Abstrat hs paper deas wth the souton of statay undetermned pane frames by usng a mxed foredspaement method based on the use of the dfferenta equatons of both the bar axa deformaton and the beam bendng. he unknowns n sovng the agebra equatons derved by the proposed approah are represented by the ntegraton onstants of eah mono-axa frame of the struture. he appatons exampes show that, even f the dmensons of the probem are arger than both ases reated to the use of the fore and of the dspaement methods, the proposed approah does not requre post-proessng for fndng any knemat and stat response quantty. herefore, ths approah an be onsdered as an aternatve to the Fnte Eement approahes for sovng pane mut-axa frames. Keywords: statay ndetermnate strutures, pane frames, beam bendng dfferenta equaton, bar axa deformaton SIGN CONVENION Fgure. Sgn onventon for: a) apped dstrbuted oads and reatons; b) nterna fores; ) dspaements.. Introduton he methods for fndng the souton, n terms of both knemat and stat quanttes, of statay ndetermnate pane frames are fundamenta tops of Strutura Mehans. In the terature, the most treated approahes are the fore and the dspaement methods [,,,4]. he use of the frst one s preferabe when the frame shows a ow number of stat ndetermnatons, whe the seond one s better f the number of deformaton modes of the frame, n a dsretzed system, s ow. When the frame s mono-axa a sutabe approah for ts souton s onsdered by the use of the bar-axa deformaton and the beam-bendng dfferenta equatons [,]. In genera, t requres the determnaton of 4+=6 ntegraton onstants that must be obtaned by mposng a orrespondng number of boundary ondtons. he poneerng works by Maauay [5], Brungraber [6], and the work by Fasone [7] generazed the use of ths approah to a the ases n whh some snguartes are present n these equatons. hey are due to some partuar oadng ondtons or to the presene of some natura and/or essenta onstrants aong the axs of the frame. In ths way the appaton of the dfferenta equaton approah an be onsdered as a genera too for sovng the mono-axa frames. No extenson of ths approah to mut-axa frames appears to be n the terature. As a onsequene, the ony true aternatve to the fore and dspaement approahes for the souton of mut-axa frames s the Fnte Eement approah. he am of ths paper s to extend the dfferenta equaton approah to the ase of mut-axa frames. In partuar, t w be shown that ths "goba" method s sutabe for a matrx notaton. As a matter of fat, eah matrx quantty an be easy expressed n terms of the geometry of the struture, the presene of the varous onstrants and the matera propertes. he number of unknowns s aways equa to 6 n, n beng the number of mono-axa frames omposng the struture. hs number s hgher than those reated to the use of the fore method, the dspaements method and the Fnte Eement approah (f the dsretzaton s not too dense). However, the proposed method has the advantage that the knowedge of the 6 n ntegraton onstants aows
76 Ameran Journa of Cv Engneerng and Arhteture mmedatey defnng any knemat and stat response, wthout the neessty of a post-proessng work, neessary, for exampe, when the Fnte Eement approah s apped.. Bar Axa and Beam Bendng Dfferenta Equatons he method presented n ths work for the exat souton of neary east statay ndetermnate frames s essentay based on the use of the assa dfferenta equatons governng the axa behavor of the bar and the defeton behavor of the beam. For ths reason these two equatons are treated n ths seton, wth further objetve to ntrodue the fundamenta symbosm used n ths work. he dfferenta equaton governng the axa generazed dspaement, u(x), of a homogeneous east bar, wth onstant axa stffness,, where E s the Young moduus and A s the ross-seton area, has the foowng form: u ( x) = px () where the apex ndates the dervatve wth respet to x and px s the axa ontnuous oad atng on the bar. It s mportant to note that, f some onentrated axa fores at on the beam, Eq. () remans vad f the generazed funtons are used as n [7]. A frst ntegraton of Eq. () eads to: () u ( x) = p ( x) + C () () ε ( x) = u ( x) = p ( x) + C where the apex nsde the brakets ndates the order of ntegraton made on the funton, ε ( x) s the generazed axa stran and C s an ntegraton onstant. In the seond of these equatons the ompatbty equaton between axa dspaement and deformaton has been taken nto onsderaton. By onsderng the onsttutve reatonshp, the axa nterna fore N( x ) an be easy found as: () N( x) = ε ( x) = p ( x) + C () A further ntegraton of Eq. () gves: () ux = p ( x) + Cx + C (4) he vaues of the onstants C and C are obtanabe by mposng the boundary ondtons that an be of stat type (based on the vaue of the nterna axa fore) and/or knemat type (based on the vaue of the axa dspaement). It s mportant to note that n both ases of statay determnate and ndetermnate bars, mposton of the boundary ondtons eads to a system of two equatons wth two unknowns admttng a unque souton. On the ontrary, when the bar s knematay ndetermnate (whh happens when the boundary ondtons are both of stat type), the agebr souton s mpossbe. he dfferenta equaton governng the defeton generazed dspaement, wx, of a homogeneous east Bernou beam, wth onstant axa stffness,, I beng the ross-seton nerta moment, has the foowng form: w ( x) = qx (5) where qx s the transversa ontnuous oad atng on the beam. he use of the generazed funton aows onsderatons of the ases of dsontnuous oads, and of presene of nterna and externa onstrans aong the axs [7]. A frst ntegraton of Eq. (5) eads to: () w ( x) = q ( x) + C (6) he foowng reatonshp hods n the ase of the Euer-Bernou beam: ( x) = w ( x) (7) ( x ) beng the nterna shear fore. Substtutng from Eq.(6) gves: () ( x) = q ( x) C (8) Whe the ntegraton of Eq. (6) gves: () w ( x) = q ( x) + Cx + C4 () M ( x) = q ( x) Cx + C4 (9) where the reatonshp between the beam bendng moment M( x) and the defeton of the Euer-Bernou beam, that s M ( x) = w ( x), has been onsdered. Further ntegraton of Eq. (9) eads to: x 4 5 () w ( x) = q ( x) + C + C x+ C () x ϕ( x) = q ( x) + C + C4x+ C5 () where the ompatbty reatonshp of the Euer-Bernou beam between the generazed rotaton, ϕ ( x) and the defeton, that s ϕ ( x) = w ( x), has been taken nto onsderaton. Fnay, the ast ntegraton of Eq. () gves: (4) x x wx = q ( x) + C + C4 + Cx 5 + C6 () 6 he four ntegraton onstants C C6 an be evauated by mposng the four boundary ondtons of the beam. In the ases of statay determnate and ndetermnate beams the agebra four equatons resutng by ths mposton defne a system admttng ony one souton. Instead, n the ase of unstabe frames the system of agebra souton s mpossbe.
When a mono-axa frame s onsdered, then ts souton an be obtaned by onsderng together, but even separatey beng ndependent, the dfferenta equaton governng the axa deformaton and the dfferenta equaton governng the beam bendng. hs means that sx ntegraton onstants must be evauated by mposng the sx boundary ondtons. he orrespondng system of sx agebra equatons gves agan a unque souton for statay determnate and ndetermnate frames, whe the souton s mpossbe when the frame s unstabe. Ameran Journa of Cv Engneerng and Arhteture 77. Mxed Fore-Dspaement Method for Frames A gener frame an be onsdered as a system of mono-axa frames onstraned externay n orrespondene of the externa nodes and nternay, eah other, n orrespondene of the nterna nodes. Fgure.a shows one of the smpest frames, whe n Fgure b a more gener frame s represented. Here, for smpty, t s assumed that a the onstrants, both externa and nterna, are fxed jonts, even f the method an be easy rearranged for other types of onstrants. Fgure. a) smpe frame; b) gener frame he gener -th mono-axa eement of a frame, referred to a oa axs system n whh the axs x ˆ ondes wth ts axs, s oaded by a gener axa dstrbuted oad, p( x ˆ ) dstrbuted oad q( x ˆ ), and by a gener transversa Fgure. mono-axa frame he behavor of ths eement s governed by the foowng two dfferenta equatons: u ( ) = px ( ˆ); ( ) w ( ) = qx ( ˆ) ( ) (, a-b) whose suessve ntegratons ead to the foowng soutons n terms of generazed dspaements: () ux ( ˆ) = p ( ) + C, + C, ; ( ) (4) ˆ ( ˆ ) ( ˆ x wx = q x) + C, + C,4 + C,5 + C,6 6 (,a-b) In order to express the boundary ondtons, t s mportant to fnd the generazed dspaements, u, w and ϕ, and the onstran reatons, R ˆx, R ŷ and M, at the extreme nodes, I and J, of the mono-axa frame, these are: ( I ) () u = ux ( ˆ) ( ˆ ) ˆ,; = = p x x = + C ( I ) (4) w = wx ( ˆ) ( ˆ ) ˆ,6; = = q x x = + C (4, a-) ( I ) () ϕ = w ( ) q ( ) = = C,5 ( ) = ˆ ˆ ( I ) Rˆ = u ( x ) ˆ ˆ,; x x= = p x x= C ( I ) () Rˆ = ( ) w ( ) ( ˆ),; y = = q x = + C ( I ) () M = ( ) w ( ) q ( ) ( ) C = = = +,4 (5, a-)
78 Ameran Journa of Cv Engneerng and Arhteture u = ux ( ˆ) = () = p ( ),,; = + C + C w = wx ( ˆ) = (4) ( ˆ = q x),,4,5,6; = + C + C + C + C 6 ϕ = w ( ) = () ( ˆ = q x) C, C,4 C,5 ( ) = ˆ ˆ = x = R u x = p( ) + C ; =, ˆ = ( ˆ ) y ˆ x= () = q = C, = ( ˆ) = R w x M w x () ; = q ( ) C C =,,4 (6, a-) (7, a-) Eah of the above mentoned four groups of reatonshps an be rewrtten n matrx form as foows: where: ( K) ˆ ( K) ( K) uˆ = A aˆ ; ( K) ˆ ( K) ˆ ( K) rˆ = B b K = I, J ( u w ϕ ) ( yˆ ) ( K) ( K) ( K) ( K) = uˆ ; ( K) ( K) ( K) ( K) rˆ = R R M K = I, J (8, a-b) (9, a-b) oets the sx ntegraton onstants of the -th mono-axa frame and ˆ ( K ) A and B ˆ K are expressed as foows: ˆ ( I ) A = ; ˆ A = 6 ( ) (, a-b) ˆ ( I ) B = ( ) ; ( ) (, a-b) ( ) ˆ B = ( ) ( ) ( ) Fnay the vetors a ˆ ( K ) and b ˆ K, whh depend on the externa oads, an be expressed as foows: () (4) I () aˆ = p ( ) ( ˆ) ( ˆ) ; = q x q x ( ) = ( ) = (, a-b) = = = () (4) () aˆ = p ( ) q ( ) q ( ) ( ) ( ) ( ) ( p ( x) = q ( x) = q ( x) = ) ( p ( x) = q ( x) = q ( x) = ) ˆ ( I ) () () () b = ˆ ˆ ˆ ; ˆ () () () b = ˆ ˆ ˆ (, a-b) When the externa nodes are fxed, the ompatbty ondtons mpy that a the dspaements are zero n these ponts. For exampe, for the frame represented n Fgure a t shoud be: ( A) ( I) ˆ ( I) ( I) uˆ uˆ = A = aˆ ( B) ( J) ˆ ( J) ( J) uˆ uˆ = A = aˆ (4, a-b) hese reatonshps represent a system of sx saar agebra equatons n the tweve unknowns defned by the omponents of the vetors and. he remanng sx equatons, neessary for sovng the system, must be reated to the ompatbty and equbrum ondtons of the free node C. In order to wrte these equatons, frsty t s neessary to refer a the stat and knemat empoyed quanttes to the same goba referene axs system. In ths operaton the ntroduton of the so-aed rotaton matrx, G, s neessary for passng from the defnton of any vetor, ˆv, referred to the oa axs system ( Ox ; ˆ, y ˆ), to the same vetor, v, referred to the goba axs system ( Oxy ;, ). If α s the ange between x ˆ and x, ths matrx has the foowng form: therefore: osα snα G = snα osα (5) v= Gv ˆ (6) he ompatbty ondtons of the free node C of the frame of Fgure a gves: ( J) ( I) ( J) ( I) = ˆ = ˆ ˆ ( J) ˆ ( I) ( J) ( I) GA GA Ga ˆ Gaˆ ( J) ( I) ( J) ( I) A A a a u u G u G u = = (7) he ast three equatons neessary for sovng the frame probem under examnaton are reated to the equbrum ondtons of the free node C: ( J) ( I) ( C) ( J) ( I) ˆ ˆ C r + r = f G r + G r = f ˆ ( J) ˆ ( I) ˆ ( I) ˆ ( J) ( C) GB + GB = Gb + Gb + f (8) ( J) ( I) ( I) ( J) ( C) B + B = b + b + f
Ameran Journa of Cv Engneerng and Arhteture 79 n whh f ( C) oets the onentrated oads apped drety on the node C. Reassumng, the sovng system of equatons s omposed by Eqs. (4), (7) and (8) and an be wrtten n the foowng ompat form: where: D = d (9) ( I ) A A D = ; ( J) ( I) A A ( J) ( I) B B ( I ) a a d = ; ( J) ( I) a a ( J) ( I) ( C) b + b + f = (, a-) It s mportant to note that n reportng the equatons n the form of Eq. (9), both the members of Eqs. (4) have been pre-mutped by the rotaton matres G and G, respetvey. Obvousy, ths ast operaton does not hange the resut. Lke the ases of the bars, the beams and the monoaxa frames, even for the ase of mut-axa frames, have ony one souton for statay determned and undetermned systems. Hene the square matrx D s nvertbe. he nverse of Eq. (9) gves the vaue of the onstant vetor and from ths vaue a the exat vaues for stat and knemat quanttes defnng the system response an be mmedatey obtaned. In a foowng seton the system represented n Fgure a w be soved as an exampe. 4. Automat Construton of the Sovng System he reatonshp gven n Eq. (9) an be extended to any gener mut-axa frame. However, the appaton of the mxed method oud be mproved f a dret onstruton of the quanttes of the Eq. (9) an be aheved. For ths purpose, t s easy to show that the onstruton of the vetor s mmedate: t s a vetor of order 6 n, f n s the number of mono-axa frames omposng the struture, oetng the ntegraton onstant vetors of eah mono-axa frame ( =,, n). he matrx D, of order 6 n 6 n, s but by matres of order 6, whe the vetor d, of order 6 n, s but by n vetors of order. Some row-boks of D refer to the externa onstraned nodes of the struture. In partuar, f the extreme pont K of the -th eement s fxed, then a row-bok of D must show a zero matres, exept for the bok orrespondng to the -th oumnbok, where the matrx A ( K ) must be nserted. It s worth rememberng that f the node onsdered s the frst one (respet to the oa axes), then K I ; otherwse, t s K J. In the orrespondng row-bok of d, the vetor ( K ) a must be nserted. Some other row-boks of D are reated to the ompatbty ondtons of the nterna free nodes: f m mono-axa frames are onneted n a node, then m row boks orrespond to these ondtons; n eah of these row-boks ony two matres are not zero, these are those orrespondng to two of the eements onneted n the node, paed n orrespondene of the oumn-boks orrespondng to the eements. If, for exampe, the monoaxa frame eements are the j-th and the k-th ones, the ( K ) ( K ) non-zero matres are A j and A k, paed n the j-th and n the k-th bok-oumn, respetvey. In orrespondene of the same row-bok, n the vetor d the vetor a ( j K) a ( K) k must be paed. he oupe of eements to be onsdered n eah of the m row-boks reated to the node n examnaton must be hosen n suh a way that no repeated ompatbtes arse (and, onsequenty, no oosng ones are). At ast, the remanng row-boks orrespond to the equbrum ondtons n the nterna free nodes. Eah row-bok orresponds to a node: f m eements are onneted n the node C, zero matres are paed n orrespondene of the oumn-boks reated to eements not onneted n C, whe the matres B ( j K ) must be paed n the ross of a the m oumn-boks orrespondng to the eements onneted n the node. In the same row-bok, n the vetor d the vetor ( C) ( K ) f + b.must be aoated m In the exampe seton the vetor and matrx quanttes neessary for budng the souton equaton of the mutaxa frames represented n Fg.b are reported. he proposed method s eary very sutabe for the mpementaton n a program auus. 5. Exampes In ths seton some exampes are gven n order to better arfy the proposed approah and ts prata appaton. 5.. Mono-axa Frame Sampes In ths sub-seton the two mono-axa frames represented n Fgure are onsdered. he expressons of the axa dspaement and of the transversa defeton are gven by the spefatons of Eqs. (4) and (): qsnα x ux = + Cx + C; qosα x x x wx = + C + C + Cx+ C 4 6 4 4 5 6 (, a-b) he frst frame of these fgures s haraterzed by the foowng boundary ondtons:
8 Ameran Journa of Cv Engneerng and Arhteture u( x) x= = C w( x) x= = C6 ϕ( x) x= = w ( x) x= = C5 qosα u( x) x= = + C + C 4 qsnα wx x= = + C + C4 + C 5 + C6 4 6 qsnα ϕ( x) x= = w ( x) x= = + C + C 4 + C5 = 6 (, a-f) ths defne a system of sx equatons n the sx unknowns C,, C6 yedng the foowng unque souton: α q os C = ; C q sn α C = ; q sn α C4 = ; C5 C6 = (, a-f) Fgure. mono-axa frame exampes: a) statay undetermned; b) unstabe At ths pont a the dspaements and nterna fores an be obtaned mmedatey. It s mportant to note that the ntegraton onstants C and C appear ony n the two Eqs. (4,a,d). hs means that they an be obtaned by sovng smpy a system of two equatons and two unknowns. he other four onstants an be obtaned by sovng the other four equatons. Obvousy ths s beause of the ndependene between the axa and the defeton probems. he seond frame onsdered n ths subseton s the unstabe one represented n Fgure b. Eqs.() are st vad, whe the orrespondng boundary ondtons are: u( x) x= = C w( x) x= = C6 M( x) x= = w ( x) x= = C4 N( x) x= = u ( x) x= = q osα + C qsnα M( x) x= = w ( x) x= = + C + C4 ( x) x= = w ( x) x= = qsnα + C = (4,a-f) ths shows an mpossbe sovng system. However, t s mportant to note that, f ony the two Eqs. (,a,d) are onsdered, they are abe to gve a souton for the onstant C and C. hs s due to the fat that the axa probem reated to the frame under examnaton s statay determned. 5.. Mut-axa Frame Sampes wo sampes are treated n ths sub-seton: the frst one s the very smpe exampe represented n Fgure a, for whh a the matrx quanttes, neessary for the souton, are gven n seton. In partuar, t s neessary to onsder the sovng equaton havng the form gven nto Eq. (9), where the matrx D and the vetors and d are gven n Eqs. (). As the node I of the eement and the node J of the eement are externay fxed, the matres ˆ ( I A ) and A ˆ J, referred to the oa referene axes, an be used ( I ) nstead of the matres A and A, respetvey. hey have the detaed expressons of Appendx (). In the same appendx the other vetors and matres used n Eqs. (9) and () are gven. Moreover, the vaues of the ntegraton are obtaned there. he vaues of these onstants gven n appendx () mpy the foowng forms of the knemat and stat aws of the two mono-axa frames: ( ˆ ) u x =.66 ; w ( ) =. + 4.8669 6 ϕ ( ) =. 4.8669 ; 5 N ( ) =.67 ( ˆ) = ( ˆ+ ) M x. x 4.8669 ; =. p osα u( ) =.958 +.5896 ; 4 p snα w( ) = 5.7 +.947 4 6 6.655 +.57 ; psnα ϕ ( ) = + 5.7 6.947 + 6.655 ; N ( ) = p osα.958 ; ( ˆ) = α + ( ˆ ) M x p sn 5.7 x.947 ; ( ) = p snα + 5.7 As sampes, n Fgure 4, Fgure 5 and Fgure 6 the defeton, bendng moment and shear fore for the eement () are shown, whe n Fgure 7, Fgure 8 and Fgure 9 the same quanttes for the eement () are shown. Fgure 4. Defeton of the eement () (m)
Ameran Journa of Cv Engneerng and Arhteture 8 Fgure 5. Bendng moment of the eement () (Nm). Fgure 6 Shear fore of the eement () (N) Fgure 9. Shear fore of the eement () (N) For the frame represented n Fg.b the unknowns are the onstants nsde the 5 vetors, wth =,,,5, oeted nto the vetor. he quanttes defnng the sovng equaton have the foowng expressons: ( I ) A ( I ) A ( I ) A 5 ( J) ( I) A A ( J) ( I) B B D = ; ( J) ( J) A A ( J) ( I) A A 4 ( J) ( J) ( I) B B B 4 ( J) ( J) A4 A5 ( J) ( J) B4 B5 (5, a) Fgure 7. Defeton of the eement () (m) Fgure 8. Bendng moment of the eement () (Nm) ( I ) a ( I ) a ( I ) a 4 ( J) ( I) a a ( J) ( I) ( A) b + b + f d = ; ( J) ( J) a a ( J) ( I) a a4 ( J) ( J) ( I) b + b + b4 ( J) ( J) a4 a5 ( J) ( J) ( C) b4 + b5 + f = 4 5 where ( A) ( C f = f ) = F. If the foowng vaues are assumed: (5, b) (5, )
8 Ameran Journa of Cv Engneerng and Arhteture 4 = 5 m, A=. m, I =.6 m, E =. N / m, q =. N / m, F = p then the foowng vaues of the ntegraton onstants are obtaned:.99 77.97 5 = ; 5.795.456 9.897 9.88 = ;.96999 4.657 89.4696.4746 9. =.9 4 5.59 87.785.697 = ;.488 9.977.695.4 7.779 = 8.547 (4a-) (4d,e) Startng from these vaues, the evauaton of any knemat or stat response of any eement omposng the frame s mmedate. Conusons A new approah for the evauaton of pane frames has been presented. It an be onsdered as a goba, or a mxed fore-dspaement method beause t takes nto aount, at the same tme, the equbrum and the ompatbty ondtons governng the strutura mehans of these systems. In partuar, t utzes the dfferenta equatons governng the probem of the monoaxa frames (bar and beam behavor) and onsders the unknowns of the probem as the ntegraton onstants of these equatons. Even f number of these unknowns s greater than the unknowns reated to the use of the fore method, those reated to the use of the dspaement method and those referrng to the appaton of a Fnte Eement approah (f a not too dense dsretzaton s used), ths drawbak s ompensated by the fat that no post-proessng s requred for evauatng any stat and knemat response of the frame. he appatons of the proposed approah to some smpe exampes has shown ts feasbty and predsposton to be mpemented n a omputer ode. he theoreta and prata matera reported n ths paper may be nterestng not ony for engneerng students, but aso for any sentst workng n the area of Strutura Mehans and nterested to the methods for sovng statay undetermned strutures. Referenes [] E.P. Popov, Engneerng Mehans of Sods, Prente-Ha (998). [] P.P. Benham and F.V. Warnok, Mehans of Sods and Strutures, Ptman Pubshng (976). [] F. Durke, W. Morgan and D.. Wams, Strutura Mehans, Longman (996). [4] R. Huse and J. Can, Strutura Mehans, Pagrave (). [5] W.H. Maauay, Note on the defeton of the beams, Messenger of Mathemats, 48, pp.9- (99). [6] R.J. Brungraber, Snguarty funtons n the souton of beamdefeton probems, Journa of Engneerng Eduaton (Mehans Dvson Buetn),.55(9), pp.78-8 (965). [7] G. Fasone, he use of generased funtons n the dsontnuous beam bendng dfferenta equatons, Internatona Journa of Engneerng Eduaton, 8(), pp.7-4 (). Appendx () In ths appendx the vetors and the matres appearng n Eqs (9) and () are gven. In partuar:
Ameran Journa of Cv Engneerng and Arhteture 8 ˆ ( I) ˆ ( J) A = ; A = 6 (6, a-b) whe the vetors ˆ I a and a ˆ J have the foowng expressons ( I ) aˆ = ; 4 ( J) ( J) aˆ = a = posα psnα psnα 4 6 he other matres nsde D are gven by: ( J) ˆ ( J) A = GA = = ˆ = = (7, a-b) (8, a) ( I) ( I) A A (8, b) B ˆ = = ( J) ( J) GB = (8, ) ( I) ˆ ( I) B = B = (8, d) he remanng vetors to be nserted nto the vetor d are: a J = Ga J = (9, a) ˆ
84 Ameran Journa of Cv Engneerng and Arhteture Hene, the sovng equatons have the form: a I I = a = (9, b) ˆ b J = Gb ˆ J = ( ) (9, ) ( I ) ˆ ( I ) ( C ) b = b = ; f = ( F M) (9, d-e) () () () 6 () 4 () 4 5 () 6 () = () () () 4 () 5 () 6 Substtutng: posα psnα 4 psnα 6 F M 4 = 5 m, A=. m, I =.6 m, E =. N / m, p =. N / m, F = p, M = p, α = 45 then the foowng vaues of the ntegraton onstants are obtaned: () () () () () 4 5 () () () () 6 =.958 ; =.5896 ; = 5.7 ; () () () C4 =.947 ; C5 = 6.655 ; C6 =.57 C =.66 ; C C =. ; C = 4.8669 ; C C C C C (4)