Analysis of Infinitely Long Euler Bernoulli Beam on Two Parameter Elastic Foundation: Case of Point Load

Transcription

1 Analysis of Infinitely Long Euler Bernoulli Beam on Two Parameter Elastic Foundation: Case of Point Load Dr. Mama, Benjamin Okwudili Department of Civil Engineering, University of Nigeria, Nsukka. Dr. Ike, Charles Chinwuba* Department of Civil Engineering, Enugu State University of Science and Technology. Dr. Nwoji, Clifford Ugochukwu Department of Civil Engineering, University of Nigeria, Nsukka. Dr. Onah, Hygynus Nwankwo Department of Civil Engineering, University of Nigeria, Nsukka ABSTRACT In this study, the flexural analysis of infinitely long Euler Bernoulli beam continuously resting on generalised two-parameter elastic foundation was performed using the method of undetermined parameters, for the case of point load acting at the origin of coordinates. Analytical solutions were obtained for the deflection, foundation reaction pressure, bending moment and shear force distributions over the longitudinal axis of the beam. Maximum values of the deflection, foundation reaction, bending moment and shear force were found to occur at the point of loading. It was shown that the solutions simplify to solutions for infinite Euler Bernoulli beams on Winkler foundation when the second foundation parameter vanishes. KEYWORDS: Euler Bernoulli beam, generalized two parameter elastic foundation, method of undetermined parameters. INTRODUCTION Background Beams on elastic foundation problems occur often in engineering applications in geotechnical, railway, highway pavements, and marine engineering. The fundamental issue in the analysis is the modelling of the contact between the structural element (beam and the soil medium as well as modelling one beam. Beam models that have been used are Euler Bernoulli beam model, Timoshenko beam and Mindlin beam

2 Vol. [07], Bund In the Euler Bernoulli beam theory, shear deformations are neglected and plane cross-sections are assumed to remain plane and normal to the longitudinal axis after deformation. Timoshenko assumed that plane cross-sections of the beam will remain plane but no longer normal to the crosssection after deformation in order to account for shear deformation which was disregarded in the Euler Bernoulli theory. Timoshenko beam theory considers the effects of shear deformation and rotational inertia in the formulation of the equation for flexure under loads. Timoshenko s equation for bending of isotropic beams of constant cross-section is given by EI 4 d w = p( x EI d p ( 4 αag where A is the area of the cross-section, G is the shear modulus, α is the shear correction factor, E is the Young s modulus of elasticity, I is the moment of inertia. w(x is the transverse deflection, p(x is the distribution of transverse load and x is the longitudinal axis of the beam. Timoshenko defined the shear correction factor as: average shear strain on a cross-section k = ( shear strain at the centroid Timoshenko beam theory is a first order shear deformation theory that predicts constant shear strain distribution through the cross-section; and is only suitable for the analysis of moderately thick beams. Elastic foundation models that have been used to model the soil include Winkler model, Pasternak model, Hetenyi model, Filonenko Borodich model, Kerr model, Reissner simplified elastic continuum model, Vlasov and Leontiev elastic continuum models; and the Elastic Continuum model. In the Winkler foundation model, the soil layer is replaced by a set of independent closely spaced elastic springs with a spring constant, k. The soil reactive pressure on the beam at any arbitrary point, x is proportional to the deflection at the point and can be expressed as p( x = kw( x (3 where k is the Winkler s coefficient of subgrade reaction at point x. Winkler foundation is a single parameter model since only one parameter, k is used to describe the soil reaction. The fundamental demerit of the model is that it differs from discontinuity of the deformations because it does not take account of shear stresses. This absence of shear coupling is the single most significant shortcoming of the Winkler model. Two-parameter foundation models account for the displacement continuity of the foundation which is the major defect of the Winkler foundation by the introduction of a second parameter. The two-parameter foundation models derived by Filonenko Borodich (945, Pasternak (954 and

3 Vol. [07], Bund Hetenyi (946 provide for the displacement continuity of the soil medium by the adding of a second spring which interacts with the first spring of the Winkler model. Kerr (964, 965 has generalized the Pasternak foundation model by the inclusion of a third spring in the vertical direction. In the Filonenko Borodich model, the second foundation parameter k is the tension T in a stretched membrane. In the Hetenyi foundation model, the second foundation parameter is the beam s stiffness where displacement continuity is introduced by adding an imaginary beam in bending. In the Pasternak foundation, displacement continuity is provided for by the introduction of a virtual shear layer which integrates the vertical spring elements and the second foundation parameter is the shear modulus G of the shear layer. (Dinev, 0. Simplified elastic continuum foundation models have been derived by Reissner (958 and Vlasov and Leontíev (966 who made simplifying assumptions to the formulation of elastic continuum foundations by introducing functions for the distribution of displacements or the stresses in the soil medium (Dinev, 0; Teodoru, 009. The soil reaction p(x for two-parameter foundation models is given in general by: ( s ( ( dwx p x = kw x k (4 where k and k are the two foundation parameters. RESEARCH AIM AND OBJECTIVES The research aim is to do a flexural analysis of infinitely long Euler Bernoulli beam resting on a generalized two-parameter elastic foundation for the case of a point load at the origin. The objectives are; i. to use the method of undetermined parameters to find the deflection and maximum deflection of infinite Euler Bernoulli beam on two parameter foundation for the case of point load applied at a point on the beam. ii. to find the bending moment distributions using the bending moment curvature relations, and the maximum bending moments. iii. to determine the shear force distributions along the Euler Bernoulli beam on two parameter foundation for point load applied at the origin, O. iv. to determine the distribution of foundation reaction, and the maximum value of the foundation reaction.

4 Vol. [07], Bund THEORETICAL FRAMEWORK The fundamental assumptions of the Euler Bernoulli beam theory used in this study are: (Xin- She Yang, 007 i. Cross-section of the beam which are plane and normal to the longitudinal axis of the beam before transverse deformation remain plane and normal to it after deformation. ii. Beam is isotropic and elastic. iii. Shear deformations are neglected and beam deformations are dominated by bending. Distortion and rotation are negligible. iv. The transverse deflections are small, and the equations of small displacement elasticity theory can be used with a constant cross-section along the longitudinal axis. v. Beam is long and slender. The Euler Bernoulli differential equation for the flexure of isotropic beams of constant cross-section is EI 4 dwx ( 4 = p( x (5 where w(x is the deflection of the neutral axis, I is the moment of inertia, E is the Young s modulus of elasticity and p(x is the applied transverse load distribution. For beam on elastic foundation, the equation is where p s is the foundation reaction. 4 dw EI p( x p ( 4 s x = (6 The derivation of Euler Bernoulli beam on elastic foundations is based on four key principles. These are kinematic relations, the stress strain law, differential equations of equilibrium and resultants. The kinematic relation for small displacement elasticity is ε xx u = (7 x where ε xx is the normal strain in the x direction, and u is the displacement in the x direction. The transverse displacement w and rotation ψ are related as uxz (, = ψ( xz (8 ε xx u ψ = = z x x (9

5 Vol. [07], Bund But or simply dw ψ = θ = (0 dw u = z ( Hence, dw ε xx = z ( Stress strain law There are no forces acting in the y direction and the beam is in a state of plane stress. The stress strain relations are: σ( xz, = Eε( xz, (3 εxx = ( σ xx µσzz (4 E εzz = ( σ zz µσxx (5 E µ εyy = ( σxx + σzz (6 E ε ε xz xy + µ = σxz (7 E = ε = 0 (8 yz σ xz ux uz w w = G G + = = 0 z x x x (9 where ε yy, ε zz are normal strains; ε xy, ε yz, ε xz are shear strains; σ xx, σ yy, σ zz are normal stresses, σ xy, σ yz, σ xz are shear stresses. The transverse normal stresses σ zz may be neglected in comparison with flexural stresses σ xx. This is similar to the assumption concerning the kinematics of deformation where the transverse normal strain was assumed negligible in comparison with the longitudinal strain. In reality the longitudinal stresses in beams are very much greater than the transverse stresses. This assumption simplifies the stress strain law to one dimension as Equation (3: Resultants The resultant moment M(x and shear force Q(x expressions are given as:

6 Vol. [07], Bund M( x = zσ( x, z dzdy (0 Ax ( σ where A(x is the integration domain. Q( x = ( x, z dzdy ( xz Equilibrium equations: The equilibrium equations are derived from a consideration of the forces on an elemental part of the beam as shown in Figure. Figure : Free body diagram of an element of the beam on elastic foundation From the free body diagram of an element of the beam shown in Figure, the shear force and bending moment values at the left hand end are Q(x and M(x and at the right hand end, they are Qx ( + and M( x+. The distributed soil reaction force is p s (x while the distributed applied load is p(x. Equilibrium of vertical forces yields Qx ( px ( + p Qx ( + = 0 ( Qx ( + Qx ( { } s 0 0 s Q dq Lt = p( x px ( = Lt = = ps( x px ( (3 Let 0, Qx ( + Qx ( Q Lt = Lt = p s( x px ( (4 0 0 dq p s px ( = (5

7 Vol. [07], Bund For moment equilibrium, x x M( x x M( x p( x x + + ps Q( x = 0 (6 ( M( x+ M( x + ( p ps Q( x = 0 (7 ( M( x+ M( x + ( p ps = Q( x (8 M( x+ M( x + ( p ps = Qx ( (9 In the limit as 0 M( x+ M( x Lt + ( p p = Qx ( (30 s 0 M dm Lt = Qx ( = (3 0 dm Qx ( = (3 d M dq( x = = ps px ( (33 The classic Euler Bernoulli beam on elastic foundation equation is obtained by elimination of the shear force from the differential equation of equilibrium; and we obtain dm d = zs( x, z dzdy = p + p s (34 Using the constitutive (stress strain law, we have dm d = zeε( x, z dzdy = p + p s (35 d E zε( x, z dzdy = p + p s (36 Using the strain displacement relations

8 Vol. [07], Bund d dw E z z dzdy p p = + s (37 dy s d E z dzdy p p = + (38 dw s d E z dzdy = p p (39 d dw E I( x p p = s Ax ( Ax ( (40 I ( x = z dzdy = z da (4 d dwx ( E I( x p( ( + s x = px (4 For Euler Bernoulli beams with prismatic cross-sections resting on elastic foundations Equation (4 becomes Equation (6. For two parameter foundations, p s is given by Equation (4. The governing differential equation of equilibrium of Euler Bernoulli beams on two parameter elastic foundation is: 4 dw 4 ( dw EI + k w k = p x (43 Governing Equation The differential equation of equilibrium for the flexure of Euler Bernoulli beam of infinite length on generalized two parameter elastic foundation is given by the fourth order equation with constant coefficients given by Equation (43; where x, is the longitudinal axis of the Euler Bernoulli beam; w(x is the transverse deflection; E is the Young s modulus of elasticity; I is the moment of inertia, p(x is the distributed transverse load; k and k are the two parameters of the elastic foundation.

9 Vol. [07], Bund RESULTS Analytical solution A closed form solution to the fourth order ordinary differential equation (ODE can be sought in the exponential form, for the homogeneous case. Thus, let the trial solution be of the form where m is an undetermined parameter. mx w( x = exp mx = e (44 By substitution into the homogeneous form of the ODE, the characteristic equation for nontrivial solutions becomes the fourth degree polynomial is in m: or, Let Then, we have 4 0 EIm k m + k = (45 4 k k m m 0 EI + EI = (46 4 k λ = (47 4EI m k m + EI 4 λ = 0 ( The solutions for m are: m k k k = ± + 4λ 4λ EI EI EI (49 The four roots of m become: m, = ± ( α iβ (50 where (5 m34, = ± ( α+ iβ (5 k α = λ + EI k β = λ EI (53 The general solution becomes: α x 3 4 wx ( = ( c cos βx+ c sin βxe + ( c cos βx+ c sin βxe (54 where c, c, c 3 and c 4 are the four constants of integration.

10 Vol. [07], Bund For bounded solution of deflection, bending moment and shear force, (w(x, M(x and Q(x, for 0 x c = 0 (55 or c = 0 (56 and the bounded solution for deflection becomes: wx ( = ( c cos βx+ c sin βxe αx 0 x ( and for x < 0 c 3 c4 0 = = (58 and wx ( = ( ccos βx+ csin βxe αx x 0 (59 The integration constants are obtained by the enforcement of boundary conditions. Analytical solutions for point load acting on an infinite Euler Bernoulli beam on two parameter foundation A point load P applied at the origin (0, 0 of coordinates of an infinitely long Euler Bernoulli beam resting on a generalised two parameter elastic foundation, as shown in Figure was considered. Figure : Euler Bernoulli beam of infinite length resting on a generalised two parameter elastic foundation The boundary conditions are given by: dw θ ( x = 0 = ( x = 0 = 0 (from symmetry (60

11 Vol. [07], Bund p( x = P (from equilibrium of applied load and soil reaction (6 or p( x = P (6 0 where px ( = kw kw ( x ( θ( x = c( αcos βx+ βsin βxe + c( αsin βx+ βcos βxe ( θ( 0 = c α+ c β = 0 (65 c 3 c4β = (66 α Also, from the equilibrium of applied load and soil reaction forces, α x α x α x α 3 β + 4 β 3 α β β + αβ β x k ( c cos xe c sin xe k c [( cos xe sin xe ] P kc 4 [( α β sin βxe αβcos βxe ] = (67 Integrating, α β ( α β α αβ kc + kc kc + α + β α + β α + β α + β ( α β β αβ P kc 4 = α + β α + β (68 Solving, c c 4 3 P( α + β = (69 4βk P( α + β = (70 4αk So,

12 Vol. [07], Bund P( α + β wx ( = ( βcos βx+ αsin βxe (7 4βk α wx ( = ( βcos βx+ αsin βxe (7 k αβ Maximum deflection The maximum deflection occurs at the point of application of the load (x = 0 and is given by: wmax = wx ( = 0 = (73 k a Bending moment distribution M(x The bending moment distribution M(x is given by: dwx ( M( x = EI (74 d M( x = EI ( βcos βx + αsin βx e (75 k αβ EI ( α + β M( x = ( βcos βx αsin βxe (76 k αβ The maximum bending moment occurs at x = 0, and is given by: M max EI ( a + β = M( x = 0 = (77 k a M max P = (78 4a

13 Vol. [07], Bund Distribution of foundation reaction p(x The elastic foundation reaction pressure is given for two parameter foundations from Equation (4 by: d kαβ k px ( = k ( βcos βx+ αsin βxe k ( βcos βx+ αsin βxe αβ 4 kp λ (79 px ( = ( βcos βx+ αsin βxe + ( βcos βx αsin βxe (80 αβ k αβ The maximum value of p(x is given by: λ k pmax = px ( = 0 = + a k (8 Shear force distribution The shear force distribution is given by: Q( x = EIw ( x (8 3 EId Qx ( = ( βcos βx+ αsin βxe (83 3 k αβ 4 EI Qx ( = (( α β sin βx αβcos βxe (84 k αβ The maximum value of Q(x occurs at x = 0, and is given by: P Q( 0 = Qx ( = 0 = = Qmax (85 Relationship with infinite Euler Bernoulli beam on Winkler foundation When k = 0, the foundation becomes a Winkler foundation, and: α = ± λ, β = ± λ (86

14 Vol. [07], Bund and wx ( = ( c3cos λx+ c4sin λxe λx (87 for 0 x for x 0 wx ( = ( c cos λx+ c sin λxe λx (88 λx wx ( = (cos λx+ sin λxe (89 k wmax = w( 0 = (90 k EI ( λ λ λx M( x = (cos λx sin λxe (9 k λ 3 EI λx M( x = (cos λx sin λxe (9 k P x M( x = (cos λx sin λxe λ (93 4λ P Mmax = M( 0 = (94 4λ p Q max max = = (95 λ P = (96 DISCUSSION In this paper, closed form analytical solutions have been obtained for the flexural problem of an infinitely long Euler Bernoulli beam continuously supported on a generalised two parameter elastic foundation for the case of point load P applied at the origin (0,0. The method of undetermined parameters was used in integrating the fourth order ordinary differential equation (ODE. A trial solution for the unknown deflection function in the ODE was assumed in the exponential function form given in Equation (44. For homogeneous equations, nontrivial solutions were obtained for the characteristic equation given as the fourth degree polynomial in Equation (48. The basis of solutions were then obtained from Equations (50 and (5. The boundedness condition was applied to obtain

15 Vol. [07], Bund bounded solutions for deflection as Equation (57 for x > 0 and Equation (59 for x < 0. The specific case of point load P at the origin was considered. The boundary conditions were obtained from the considerations of the symmetrical form of the problem and the equilibrium of applied point load and the foundation reaction; and are given as Equations (60 and (6. The boundary conditions were applied to obtain the unknown integration constants as Equations (69 and (70 and deflection as Equation (7. Maximum deflection was observed to occur at the point of loading and is given as Equation (73. The bending moment deflection relation for Euler Bernoulli beams was used to find the bending moment distribution as Equation (76. Similarly, the shear force was obtained using the shear force deflection equation as Equation (84. Maximum bending moments and shear force values were found to occur at the point of load and were found as Equations (78 and (85. The distribution of foundation reaction was also computed and found as Equation (80. It was observed that if k = 0, the deflection, bending moment and shear force distributions become the same as the deflection, bending moment and shear force distributions for infinite Euler Bernoulli beam on Winkler foundation. The following conditions can be made: CONCLUSIONS (i The method of undetermined parameters has yielded mathematically closed form solutions to the flexural analysis of infinite length Euler Bernoulli beam continuously supplied on generalised two parameter elastic foundation. (ii The solutions obtained for the deflections, bending moments, shear force distributions and the foundation reaction are exact solutions within the limitations of the foundational Euler Bernoulli theory of beams and the elastic foundation model assumed. (iii When the second foundation parameter k is made to vanish in the resulting equations, the solutions obtained become identical with the corresponding solutions for Euler Bernoulli beam resting on Winkler foundation for the case of point load at the origin. REFERENCES [] Dinev, D. (0: Analytical Solution of Beam on Elastic Foundation by Singularity Functions. Engineering Mechanics Vol, 0, No. 6 pp [] Filonenko Borodich M.M. (945: A very simple model of an elastic foundation capable of spreading the load. Shornite Moskovkovo Elektro Instituta. [3] Hetenyi, M. (946: Beams on Elastic Foundation: Theory with Applications in the Fields of Civil and Mechanical Engineering. The University of Michigan Press, Ann Arbor, Michigan. [4] Kerr, A.D. (964: Elastic and visco-elastic foundation models. J. Appl. Mech 3: pp

16 Vol. [07], Bund [5] Kerr, A.D. (965: A study of a new foundation model. Acta Mechanica Vol. pp [6] Myslecki, K. (004: Approximate fundamental solutions of equilibrium equations for thin plates on elastic foundation. Archives of Civil and Mechanical Engineering Vol IV, 004, No. pp [7] Pasternak, P.L. (954: On a new method for analysis of an elastic foundation by means of two foundation constants (in Russian Gosud. Izd. Lit. po Stroitelstvu Arkhitekture, Moscow. [8] Reissner, E. (958: Deflection of plates on viscoelastic foundation. J. of Appl. Mech. Vol 5. pp [9] Teodoru, I.B. (009: Beams on elastic foundation the simplified continuum approach. Buletinul Institutulni Politechnic lin Lasi Tomul LV (LIX Fasc4, 009. [0] Vlasov, V.Z. and Leontiev, N.N. (966: Beams, plates and shells on elastic foundations. Translated from Russian by Israel Program for Scientific Translations Accession No N [] Xin She Yang (007: Applied Engineering Mathematics. Cambridge International Science Publishing (CISP Cambridge UK p.96. [] Zhan Yun-gang: Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method Electronic Journal of Geotechnical Engineering, 005 (Vol.0, pp Available at ejge.com. 07 ejge Editor s note. This paper may be referred to, in other articles, as: Dr. Mama Benjamin Okwudili, Dr. Ike Charles Chinwuba*, Dr. Nwoji Clifford Ugochukwu, Dr. Onah Hyginus Nwankwo: Analysis of Infinitely Long Euler Bernoulli Beam on Two Parameter Elastic Foundation: Case of Point Load Electronic Journal of Geotechnical Engineering, 07 (.3, pp Available at ejge.com.