ANALYSIS OF TIMOSHENKO BEAM RESTING ON NONLINEAR COMPRESSIONAL AND FRICTIONAL WINKLER FOUNDATION

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1 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. ANALYSIS OF TIMOSHENKO BEAM RESTING ON NONLINEAR COMPRESSIONAL AND FRICTIONAL WINKLER FOUNDATION Adel A. Al-Azzaw Department of Cvl Engneerng, Nahran Unversty, Baghdad, Iraq E-Mal: ABSTRACT Ths paper deals wth lnear elastc behavor of deep beams restng on lnear and nonlnear Wnkler type elastc foundatons wth both compress onal and tangental resstances. The basc or governng equatons of beams on nonlnear elastc Wnkler foundaton are solved by fnte dfference method. The fnte element method n Cartesan coordnates s formulated usng two dmensonal plane stress soparametrc fnte elements to model the deep beam and elastc sprngs to model the foundaton. Two computer programs coded n fortran_77 for the analyss of beams on nonlnear elastc foundatons are developed. Comparsons between the two methods and other studes are performed to check the accuracy of the solutons. Good agreement was found between the solutons wth percentage dfference of 3%. Several mportant parameters are ncorporated n the analyss, namely, the vertcal subgrade reacton, horzontal subgrade reacton and beam depth to trace ther effects on deflectons, bendng moments and shear forces. Keywords: beams, Wnkler foundaton, fnte dfference, fnte elements, frcton, nonlnear. INTRODUCTION In the analyss of elastcally supported beams, the elastc support s provded by a load-bearng medum, referred to as the foundaton along the length of the beam. Such condtons of support can be found n a large varety of geotechncal problems. There are two basc types characterzed by the fact that the pressure n the foundaton s proportonal at every pont to the deflecton occurrng at that pont and s ndependent of pressure or deflecton produced at other pont (Fgure-). The second type s furnshed by elastc sold, whch n contrast to the frst one represents the case of complete contnuty n the supportng medum. Fgure-. Smply supported beam under external load and foundaton resstances. Al-Hachm (997) presented a theoretcal analyss for predctng the large dsplacement elastc stablty analyss of plane and space structures subjected to general statc loadng. The beam-column theory was used n ths analyss, takng nto accounts both bowng and axal force effects. The general equatons of fxed end moments of a beam subjected to lateral loads were also derved. The work employed ths analyss to study the behavor of beams wth elastc foundatons, ples drven nto sol and large dsplacements of submarne ppelnes. Onu () derved a formulaton leadng to an explct free-of meshng stffness matrx for a beam fnte element foundaton model. The shear deformaton contrbuton was consdered and the formulaton was based on exact soluton of the governng dfferental equaton. Arstzabal-Ochoa () developed, n a smplfed manner, a nonlnear large deflecton-small stran analyss of a slender beam-column of symmetrcal cross secton wth sem rgd connectons under end loads (conservatve and no conservatve), ncludng the effects of axal load eccentrctes and out-of-plumpness. Guo and Wetsman () made an analytcal method, accompaned by a numercal scheme, to evaluate the response of beams on no unform elastc formulaton, where the foundaton modulus s K z = K z (x). The method employed Green s foundaton formulaton, whch results n a system of nonsngular ntegral equatons for the dstrbuted reacton p(x). Lazem (3) presented a theoretcal analyss for large dsplacement elastc stablty of n-plane structures where some members were embedded nto or restng on elastc foundatons. The analyss was based on Euleran formulaton, whch was developed ntally for elastc structures and was extended to nclude sol-structure nteracton. Al-Azzaw and Al-An (4) studed the lnear elastc behavor of thn or shallow beams on Wnkler foundatons wth both normal and tangentonal frctonal resstances. The fnte dfference method was used to solve the governng dfferental equatons and good results were obtaned wth the exact solutons for dfferent load cases and boundary condtons. Al-Musaw (5) studed the lnear elastc behavor of deep beams restng on elastc foundatons. The fnte element method n Cartesan coordnates s formulated usng dfferent types of one, two and three dmensonal soparametrc elements to compare and check the accuracy of the solutons.

2 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. Al-Azzaw () used the fnte dfference method for solvng the basc dfferental equaton for the elastc deformaton of a thn beam supported on a nonlnear elastc foundaton. A tangent approach was used to determne the modulus of subgrade reacton after constructng a second degree equaton for load-deflecton dagram. Results of plate loadng test of sol obtaned n Iraq were used n the analyss. An teratve approach s used for solvng the nonlnear problem untl the convergence of the soluton. Al-Azzaw and Theeban () studed the geometrc nonlnear behavor of beams restng on Wnkler foundaton. Tmoshenko s deep beam theory s extended to nclude the effect of large deflecton theory usng fnte dfferences. In the fnte element method (ANSYS program), the element SHELL 43was used to model the beam. Al-Azzaw, Mahdy and Farhan () studed the nonlnear materal and geometrc behavors of renforced concrete deep beams restng on lnear and nonlnear Wnkler foundatons. The fnte elements through ANSYS (Release-, 7) computer software were used. The renforced concrete deep beam s molded usng (SOLID 65) 8 node brck element and the sol s molded usng lnear sprng (COMBIN 4) element or usng nonlnear Wnkler sprng (COMBIN39) element. ELASTIC FOUNDATION Wnkler model for both compressonal and frctonal resstances are used to model the elastc foundatons. Ths model assumes that the base s consstng of closely spaced ndependent lnear sprngs, consequently as shown n Fgure-. Fgure-. Wnkler compress onal and frctonal model. Modulus of subgrade reacton s a conceptual relatonshp between sol pressure and deflecton. It can be measured by usng plate-loadng test. Usng ths test, a load-deflecton curve s adopted. The modulus of subgrade reacton K z can be calculated usng: p K z = () w where: K z p w s the modulus of subgrade reacton, s the appled pressure and s the deflecton. The value of K z s obtaned from the concept of tangent approach as shown n Fgure-3. Fgure-3. Typcal sol pressure-settlement curve.

3 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. There are a wde range of K z values for dfferent types of sol. In the present study, a quas lnearzaton method or teraton procedure to get the value of K z s used. Ths lnearazaton by teraton method was developed usng the tangent method as a basc approach. In ths study, the lnear and nonlnear behavors are adopted. The nonlnear behavor s modeled usng teratve values of K z. A typcal p-w dagram was taken from a plate loadng test whch was carred out on a sol n Baghdad. The result of ths test s shown n Fgure-4. The Consultant Engneerng Bureau n the Unversty of Baghdad had carred out ths test n Al-Muthana arport regon for the Bg Baghdad Mosque project. Load (kpa) Settlement (mm) Fgure-4. Plate loadng test data [consultant engneerng bureau / unversty of Baghdad]. The data shown n the load-deflecton curve s used to obtan the followng second degree polynomal equaton: K z (w) 7 9 = * w 8.* w () whch gves, the ntal modulus of subgrade reacton= 8 kn/m 3 and the fnal modulus of subgrade reacton = 749 kn/m 3 for w.5m. In the present study, the horzontal subgrade reacton s assumed to have the same values and behavor of vertcal subgrade reacton. ASSUMPTIONS AND GOVERNING EQUATIONS FOR DDEEP BEAMS The man assumptons are: a) Plane cross sectons before bendng reman plane after bendng. b) The cross secton wll have addtonal rotaton due to transverse shear. Warpng of the cross secton by transverse shear wll be taken nto consderaton by ntroducng a shear correcton factor (c ). The governng equatons of deep beams on elastc foundatons characterzed by Wnkler model for compressonal and frctonal resstances could be obtaned [deep beam wth unform subgrade by Al-Jubor (99)]: d d w Gc A( + ) + K z ( w) w = q dx dx ψ (3) d ψ dw h EI Gc A ψ + = K ( w) ψ (4) x dx dx where G s the shear modulus, c s the shear correcton factor (c =5/6 for rectangular cross sectons and c = for I-sectons), A s the cross-sectonal area of the beam, ψ s the rotaton of the transverse sectons n xz-plane of the beam, w s the transverse deflecton, E s the modulus of elastcty of the beam materal, q s the transverse load per unt length, K and K are the lnear or nonlnear z x modul of subgrade reacton n z and x drectons and I s the moment of nerta of the beam secton. In case of the depth of the beam decreases (thn beam) the shear modulus becomes nfnte and equaton 3 vanshes as dw ψ = and equaton reduces to the case of small dx deflecton of thn beams. FINITE DIFFERENCE METHOD The fnte dfference method s one of the most general numercal technques. In applyng ths method, the dervatves n the governng dfferental equatons under consderaton are replaced by dfferences at selected ponts. These ponts or nodes are makng the fnte dfference mesh. In the analyss of deep beams by ths method, the dfferental equatons at each pont (or node) are replaced by dfference equatons. By assemblng the dfference equatons for all nodes, a number of smultaneous algebrac equatons are obtaned and solved by Gauss-Jordan method.

4 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. The beam s dvded nto ntervals of ( x) n the (x) drecton as shown n Fgure-5, assumng (n) to represent the number of nodes and () the node number under consderaton. In the fnte dfference method, the curve profle of the beam deflecton s approxmated by a straght lne between nodes for the fnte dfference expressons of the frst dervatves and by a parabola for the second dervatves. Fgure-6. Plane stress element. Fgure-5. Fnte dfference mesh for the deep beam. The governng equatons are rewrtten n fnte dfferences and are produced for an nteror node (): ψ+ ψ Gc A [ x ψ EI[ - + w + ( w + w ( x) - )] + q K (w ).w = z (5) + ψ + ψ- w+ w- h (6) ] Gc A[ψ + ( )] = [ Kx = Kz( w )] ψ x x The soluton of the governng dfferental equatons of deep beams must smultaneously satsfy the dfferental equatons and the boundary condtons for any gven beam problem. Boundary condtons are represented n fnte dfference form by replacng the dervatves n the mathematcal expressons of varous boundary condtons by ther fnte dfference approxmatons. When central dfferences are used at the boundary nodes, fcttous ponts outsde the beam are requred. These may be defned n terms of the nsde ponts when the behavor of the beam functons are known at the boundary nodes. FINITE ELEMENT ANALYSIS The fnte element method s an approxmate method for the analyss of framed and contnuum structures. The basc phlosophy of ths method s that the structures or the contnuum s dvded nto small elements of varous shapes and types, whch are assembled together to form an approxmate mathematcal model. In ths paper, the fnte element method n Cartesan coordnate s used to solve the problems of deep beams restng on Wnkler type elastc foundatons wth both normal and frctonal restrants. The two dmensonal soparametrc plane stress elements are used, each node have two degrees of freedom (the deflecton w and dsplacement u) as shown n Fgure-6. The two dmensonal element n local coordnates ξ and η has eght nodes as shown n Fgure-6 (Hnton and Owen 977). Each node n a plane stress element has two degrees of freedom. They are u and w. Thus, the element degrees of freedom may be lsted n the vector (or row matrx): { δ } = [, u,..., w u ] T e w 8, 8 Shape functons For the four-node soparametrc quadrlateral element, the shape functons are: N = (- ξ) (- η)/4, N = (+ξ) (- η)/4 N 3 = (+ξ) (+η)/4 andn 4 = (- ξ) (+η)/4 (7) For the eght-node soparametrc quadrlateral element, the shape functons are: N =(-ξ)(-η)(+ξ+η)/4 N =(-ξ )( -η)/ N 3 =(+ξ)(-η)(ξ-η-)/4 N 4 =(+ξ)(-η )/ N 5 =(+ξ)(+η)(ξ-η-)/4 N 6 =(-ξ )( +η)/ N 7 =(+ξ)(+η)(-ξ+η-)/4 N 8 =(-ξ)(-η )/ (8) Thus, the degrees of freedom w and u can be defned n terms of the shape functons: n w( ξ, η) = N w (9a) = n u( ξ, η) = N u (9b) = 3

5 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. The x and y coordnates can be defned n the same manner: n x( ξ, η) = N x () = n y( ξ, η) = N y () = Thus, the geometry and the assumed dsplacement feld are descrbed n a smlar fashon usng the shape functons and the model values (thus, the name of soparametrc element s gven). Jacoban matrx-[j] The Jacoban matrx [J] s obtaned from the followng expresson: ξ [J] = η y N n x ξ ξ = y = N x η η N y ξ N y η () The nverse of the Jacoban matrx [J] - can be readly obtaned usng standard matrx nverson technque: ξ η y y η ξ = (3) [J] ξ η = det J x x y y η ξ The shape functon dervatves are calculated from the expressons as: N N ξ N η = + ξ η N N ξ N η = + y ξ y η y where ξ η ξ,, and y η and obtaned from [J] -. y (4) Stran matrx-[b] The strans are defned n terms of the nodal dsplacements and shape functon dervatves by the expresson: ε ε γ x z xz N = z N N N z w u (5) The stran matrx [B ] contans shape functon dervatves whch may be calculated from the expresson (4) and the coordnates x and y whch may be calculated at the Gauss pont coordnates from the expressons () and (). Matrx of elastc constants-[d] The generalzed stress stran relatonshp for a beam of sotropc elastc materal may be wrtten as: σ σ τ or x z xz E ( ν ) νe = ( ν ) νe ( ν ) E ( ν ) ε ε E γ ( + ν) x z xz (6) { σ e } = [ D]{ ε e } (7) where {σ e } s the element stresses and [D] s the matrx of elastc constants for the sotropc elastc materal. Stress matrx-[bd] Smlarly, the stress at any pont wthn the element for a beam of sotropc materal can be expressed as: { σ } [ D][ B]{ δ } = [ S]{ δ } e = (8) e e Stffness matrx for the plane stress element-[kp] The stffness matrx for a beam wth sotropc elastc materal s gven as: + + = n T [ K p ] [B ξ η ].[D][B ].det Jd d (9) = where [D] s gven n equaton (6) for sotropc elastc martal. Numercal ntegraton can be used to evaluate the above ntegraton usng Gauss-Legendre quadrature rule. Stffness matrx for the foundaton-[k f ] For a foundaton represented by Wnkler model for both compressonal and frctonal resstances, the stffness matrx s: [R w ] [Kf ] = where [R w ] f [ R ] = W Here, K K f [R w ] [R w ] [R w ] nxn () 4

6 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. K K = + / / f N.K ξ z.detj.d. = + / / f N.K ξ x.detj.d. where (a) (b) are the shape functons for the two or three node one dmensonal beam element, matrx for the one dmensonal beam element and K x / N s the Jacoban and are the modul of subgrade reactons n z and x drectons. J / K z Computer program In the present study, the computer program (DBNEF) (Deep Beams on Nonlnear Elastc Foundatons) was formed. The program s developed to deal wth any boundary condton, such as smply supported, fxed, and free-ends deep beams. In the program, the soluton s dvded nto two steps. The frst step s to fnd the governng equatons of deflectons and rotatons and then these governng dfferental equatons for deep beams on elastc foundaton (n terms of w and ψ ) are converted nto fnte dfferences. After wrtng the fnte dfference equatons for each boundary and nteror nodes, assemblng for these equatons must be made to form a system of smultaneous algebrac equatons. Gauss-Jordan method s used n the program to solve the system of equatons to obtan deflectons and rotatons at each node. The obtaned deflectons are compared wth deflectons of the prevous teraton after changng the value of subgrade reacton and the procedure s repeated untl convergence s obtaned. The second step s to fnd the moment and shear at each node. Also, the computer program presented by Hnton and Owen (977) s modfed n ths work to be capable of solvng the problem of beams on nonlnear elastc foundatons. Two-dmensonal plane stress elements restng on Wnkler type compress onal and frctonal foundatons have been used. The numercal results obtaned from the fnte dfference and fnte element methods have been compared wth avalable exact and other analytcal and numercal results. APPLICATIONS AND DISCUSSIONS Verfcatons Smply supported beam on lnear compressonal Wnkler foundaton The problem of smply supported beam restng on lnear Wnkler foundaton solved by Al-Azzaw & Al-An, (4) usng fnte dfference method and analytcal soluton by Heteny, (974) (thn beam theory) s consdered. The same applcaton s solved by usng ten eght node plane stress fnte elements and fnte dfferences (deep beam theory). All nformaton and fnte element mesh for plane stress elements over half of the beam s shown n Fgure-7. Fgure-7. Beam on Wnkler foundaton and fnte element mesh over half of the beam. Fgures 8, 9 and show the deflecton profles, bendng moment and shear force dagrams n x-drecton for numercal, exact and the present study (fnte dfference and fnte element methods). The results show good agreements by the used methods wth percentage dfference of 3%. 5

7 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. Dstance (m) Deflecton (mm) Exact soluton Al-Azzaw and Al-An (4) Fnte element soluton Fntel dfference soluton Fgure-8. Deflecton curves for smply supported beam restng on Wnkler foundaton. 3 5 Bendng Moment (kn.m) 5 5 Exact soluton Al-Azzaw and Al-An (4) Fnte dfference soluton Dstance (m) Fgure-9. Bendng moment curves for smply supported beam restng on Wnkler foundaton. Shear (kn) Exact soluton Al-Azzaw and Al-An (4) Fnte dfference soluton Dstance (m) Fgure-. Shear force curves for smply supported beam restng on Wnkler foundaton. 6

8 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. Smply supported beam on nonlnear compressonal Wnkler foundaton The problem of smply supported beam restng on nonlnear Wnkler foundaton solved by Al-Azzaw, () usng fnte dfference method (thn beam theory) s consdered. The problem s solved by usng ten eght node plane stress fnte elements and fnte dfferences (deep beam theory). All nformaton and fnte element mesh for plane stress elements over half of the beam s shown n Fgure-. Fgure-. Beam on Wnkler foundaton and fnte element mesh over half of the beam. Fgure- shows the deflecton profle along x- drecton for the nonlnear elastc Wnkler foundaton whle Fgures 3 and 4 show the bendng moment and shearng force along x-drecton. The results show good agreement for the dfferent solutons wth percentage dfference of 3%. Deflecton (mm) Bendng moment (kn.m) Dstance (m) Al-Azzaw () Fnte element soluton Fnte dfference soluton Fgure-. Deflecton curves for smply supported beam restng on nonlnear Wnkler foundaton Dstance (m) Al-Azzaw () Fnte dfference soluton Fgure-3. Bendng moment curves for smply supported beam restng on nonlnear Wnkler foundaton. 7

9 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. Shear Force (kn) Al-Azzaw () Fnte dfference soluton Dstance (m) Fgure-4. Shear force curves for smply supported beam restng on nonlnear Wnkler foundaton. 5 Free ends beam on nonlnear compressonal Wnkler foundaton wth end load A beam of (E=5 6 kn/m, ν =.5) and havng a length of (.8m), wdth (b=.m), depth (h=.45m) and subjected to a concentrated load (P=.5 kn), s consdered as shown n Fgure-5. The beam s restng on nonlnear compressonal Wnkler foundaton wth ntal modulus (Kz=.kN/m 3 ) and ths value and other values are obtaned from plate-load test. Ths case was analyzed by Al-Hachm (997) by usng the beamcolumn method. In the present study, the fnte-element method s used to solve ths problem. The present study results of deflectons are plotted together wth Al-Hachm (997) results as shown n Fgure-6. The comparsons between the two solutons show good agreement wth percentage dfference of 3%. Fgure-5. End loaded free-ends deep beam restng on nonlnear compressonal Wnkler foundaton. Deflecton (mm) Dstance (m) Al-Hachm (997) Fnte element soluton Fgure-6. Deflecton curves for free ends beam restng on nonlnear compressonal Wnkler foundaton. Parametrc study Lnear compressonal and frctonal Wnkler foundaton The same smply supported beam wth same propertes shown n Fgure-7 s consdered here. Fgure-7 shows that the md span deflecton decreases as the depth of the beam ncreases because the stffness of the beam ncreases. Fgure-8 shows that the md span moment ncreases as the depth of the beam ncreases because the stffness of the beam ncreases. Also, Fgure-9 shows that the maxmum shear ncreases as the depth of the beam ncreases. 8

10 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. Deflecton (mm) Beam Depth (m) Fgure-7. Effect of beam depth on md span deflecton for smply supported beam under unform load (K x = and K z = kn/m 3 ). Md-Span Bendng Moment (kn.m) Beam Depth (m) Fgure-8. Effect of beam depth on md span moment for smply supported beam under unform load (K x = and K z = kn/m 3 ). 7 Maxmum Shear (kn) Beam Depth (m) Fgure-9. Effect of beam depth on maxmum shear for smply supported beam under unform load (K x = and K z = kn/m 3 ). Fgures, and show that as the subgrade reacton coeffcent ncreased the md span deflecton, md span moment and maxmum shear decreased because the stffness of the foundaton ncreases. 9

11 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. Md-Span Deflecton (mm) Vertcal Subgrad Reacton kn/m3 Fgure-. Effect of vertcal subgrade reacton on md-span deflecton for smply supported beam under unform load (K x = and beam depth=.5m). Md-Span Bendng Moment (kn.m) Vertcal Subgrade Reacton kn/m3 Fgure-. Effect of vertcal subgrade reacton on md-span moment for smply supported beam under unform load (K x = and beam depth=.5m). Maxmum Shear (kn) Vertcal Subgrade Reacton( kn/m3) Fgure-. Effect of vertcal subgrade reacton on maxmum shear for smply supported beam under unform load (K x = and beam depth=.5). Fgure 3, 4 and 5 show that as the horzontal subgrade reacton coeffcent ncreased the md span deflecton, md span moment and maxmum shear decreased because the stffness of the foundaton ncreases.

12 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. Md-Span Deflecton (mm) Horzontal Subgrade Reacton (kn/m3) Fgure-3. Effect of horzontal subgrade reacton on md span deflecton for smply supported beam under unform load (K z = kn/m 3 and beam depth=.5m). Md-Span Moment (kn.m) Horzontal Subgrade Reacton (kn/m3) Fgure-4. Effect of horzontal subgrade reacton on md span moment for smply supported beam under unform load (K z = kn/m 3 and beam depth=.5m). 5.8 Maxmum Shear (kn) Horzontal Subgrade Reacton (kn/m3) Fgure-5. Effect of horzontal subgrade reacton on maxmum shear for smply supported beam under unform load (K z = kn/m 3 and beam depth=.5m). Nonlnear compressonal and frctonal Wnkler foundaton The same smply supported beam wth same propertes shown n Fgure- s consdered. Fgure-6 shows that the md-span deflecton for the lnear and nonlnear modulus decreases as the depth of the beam ncreases because the secton flexural rgdty EI of the beam ncreases for both lnear and nonlnear foundatons.

13 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. Md-Span Deflecton (mm) 7 Lnear Wnkler foundaton 6 Nonlnear Wnkler foundaton Beam Depth (m) Fgure-6. Effect of beam depth on md-span deflecton. Fgures 7 and 8 show that the md-span moment and maxmum shear force ncreases as the depth of the beam ncreases also, because the secton flexural rgdty EI of the beam ncreases for both lnear and nonlnear foundatons. Bendng Moment (kn.m) 7 Lnear Wnkler foundaton 6 Nonlnear Wnkler foundaton Beam depth (m) Fgure-7. Effect of beam depth on md-span bendng moment. Maxmum Shear Force (kn) Lnear Wnkler foundaton 5 Nonlnear Wnkler foundaton Beam Depth (m) Fgure-8. Effect of beam depth on maxmum shear force. CONCLUSIONS From ths study, the man conclusons are gven below: a) The results obtaned from the exact, fnte dfference and fnte element solutons check the accuracy of the method used n ths paper n whch they are n good agreement. b) The effect of beam depth s sgnfcant on the results and ncreasng beam depth wll decrease the md span deflecton and ncrease the moment and shear resstances. c) The effect of frcton at the beam-foundaton nterface s found to be small on the deflecton, moment and shear. d) The effect of varyng vertcal modulus of elastc foundaton on deflecton, moment and shear s sgnfcant. e) The obtaned results show dfferent values for both deflecton and bendng moment but rather close values

14 VOL. 6, NO., NOVEMBER ISSN Asan Research Publshng Network (ARPN). All rghts reserved. for shearng force for hgh values of appled loads on the beam, whch s restng on lnear or nonlnear elastc Wnkler foundaton. The nonlnear behavor of sol was obtaned by usng hgh-appled loads (to make the dfference n results much obvous). Ths study shows that the elastc method for analyzng beam restng on Wnkler foundaton s stll vald for ordnary appled loadng on beams. The effect of beam depth on maxmum beam deflecton and bendng moment s found to be sgnfcant but not much on shearng force for nonlnear foundatons. REFERENCES Al-Azzaw A. A. and Al-An M. A. 4. Fnte dfference analyss of beams on Wnkler foundatons. Engneerng and Technology Journal, Unversty of Technology, Baghdad, Iraq. 3: Engneerng Mechancs, Amercan Socety of Cvl Engneers. 8: Heteny M Beams on Elastc Foundatons. Ann Arbor, the Unversty of Mchgan press. Hnton E. and Owen D. R. J Fnte Element Programmng. Academc Press, London, UK. Lazem A. N. 3. Large Dsplacement Elastc Stablty Analyss of Elastc Framed Structures Restng on Elastc Foundaton. M. Sc. Thess, Unversty of Technology, Baghdad, Iraq. Onu G.. Shear effect n beam fnte element on twoparameter elastc foundaton. Journal of Engneerng Mechancs, September. pp Al-Azzaw A. A.. Fnte dfference analyss of thn beams on nonlnear Wnkler foundaton. Journal of Engneerng, Unversty of Baghdad, Baghdad, Iraq. 6: -3. Al-Azzaw A. A. and Theeban D. M.. Large deflectons of thn and deep beams on elastc foundatons. Journal of the Serban Socety for Computatonal Mechancs. 4(): 88-. Al-Azzaw A. A., Mahd A. and Farhan O. Sh.. Fnte element analyss of deep beams on nonlnear elastc foundatons. Journal of the Serban Socety for Computatonal Mechancs. 4(): 3-4. Al-Hachm E. K Large Dsplacement Analyss of Structures wth Applcatons to Ples and Submarne Ppelnes. Ph.D., Thess, Unversty of Technology, Baghdad, Iraq. Al-Jubory A. A. 99. Deep Beams and Thck Plates under Generalzed Loadng. M. Sc. Thess, Faculty of Engneerng, Nahran Unversty, Baghdad, Iraq. Al-Musaw A. N. 5. Three Dmensonal Fnte Element Analyss of Beams on Elastc Foundaton. M. Sc. Thess, Faculty of Engneerng, Nahran Unversty, Baghdad, Iraq. Arstzabal-Ochoa J. D.. Nonlnear large deflectonsmall stran elastc analyss of beam-column wth semrgd connectons. Journal of Structural Engneerng, Amercan Socety of Cvl Engneers. 7: Bowels J.E Foundaton Analyss and Desgn. McGraw-Hll Company. Guo Y. J. and Wetsman Y. J.. Soluton method for beams on non-unform elastc foundatons. Journal of 3