EXPONENTIAL FOURIER INTEGRAL TRANSFORM METHOD FOR STRESS ANALYSIS OF BOUNDARY LOAD ON SOIL

Transcription

1 Tome XVI [18] Fascicule 3 [August] 1. Charles Chinwuba IKE EXPONENTIAL FOURIER INTEGRAL TRANSFORM METHOD FOR STRESS ANALYSIS OF BOUNDARY LOAD ON SOIL 1. Department of Civil Engineering, Enugu State University of Science & Technology, Enugu State, NIGERIA Abstract: The exponential Fourier transform metho has been use in this work to solve the theory of elasticity problem of an elastic half plane subject to loa applie on the bounary. The elastic half plane meium was consiere to be soil an assume to be linear elastic, isotropic, homogeneous an of semi-infinite extent ( x, ) on the coorinate plane. The problem was solve generally for any istribution of bounary loa. Specific cases of vertical an horiontal point loa at the origin, an line loa of constant intensity on one sie of the x axis were also consiere an solve by the exponential Fourier transform metho. The solutions obtaine were foun to be ientical with solutions in the technical literature obtaine by other researchers who use ifferent methos. Keywors: Exponential Fourier transforms metho, ifferential equation of equilibrium, elastic half plane problem 1. INTRODUCTION Many geotechnical engineering problems require a stuy of the transmission an istribution of stresses in large an extensive masses of soil. Some examples are wheel loas acting on embankments an their transmission to culverts, stresses from isolate footings transmitte to retaining walls an stresses in soil ue to footings [1,, 3]. The etermination of the stress istribution of any point in a soil mass ue to external loaing is important in the analysis of settlements of builings, briges an embankments. The etermination of stress fiels an isplacement fiels in soil masses of semi-infinite extent moele as linear elastic materials belong to the classical theory of elasticity []. Consequently, they are governe by the ifferential equations of equilibrium, the material constitutive laws an the kinematic relations, together with the bounary conitions impose by the loas [5, 6]. Two basic methos use in the formulation of classical problems of elasticity are the stress metho of formulation an the isplacement metho of formulation. In the isplacement formulation metho, the equations are expresse with isplacements as the primary unknowns. This reuces the number of governing equations from fifteen for a three imensional formulation to three equations. In stress formulation, the governing equations are expresse in terms of unknown stresses as the primary variables. This also reuces the number of equations to be solve. A mixe formulation is also possible, where the governing equations are expresse in terms of the isplacements an stresses as the unknown variables to be foun. The stuy uses the governing equations of the two imensional problem of elasticity for a semi-infinite linear elastic soil in the plane where x, where the bounary surface is subjecte to istribute loa. The exponential Fourier transform metho is applie in a first principles manner to the governing equations, in orer to generate solutions for a general pattern of istribute bounary loa. Specific cases of point loa an uniformly istribute line loa were also stuie. Mathematical expression for the stress fiels ue to the line loas of infinite extent acting vertically on the surface of a semi-infinite soil have been obtaine by Flammant an presente in the technical literature [7, 8, 9]. Analytical expressions for stress fiels in soil masses assume as linear elastic, isotropic, homogeneous, semi-infinite half space can also be obtaine by using Boussinesq s solutions for the point loa applie on the surface of a semi-infinite linear elastic soil as Green s functions. Solutions for stress fiels in semi-infinite elastic soil meia originally publishe by Cummings [1] an Gray [11] have been presente by Timoshenko [6], Newmark [1], Hall [13] an Forster an Fergus [1].. RESEARCH AIM AND OBJECTIVES The research aim is to use the exponential Fourier transform metho to solve two imensional elasticity problems of half space soil meia. The objectives are: (i) to use the exponential Fourier transform metho to obtain general solutions for the stresses in semiinfinite linear elastic isotropic soil meia uner bounary line loas. (ii) to etermine specific solutions for stress fiels in semi-infinite linear elastic homogeneous soil masses ue to line loas of constant intensity acting along the x axis. (iii) to etermine specific solutions for stresses ue to vertical an horiontal point loas applie at the origin on the surface of the elastic half plane. 13 F ascicule 3

2 Tome XVI [18] Fascicule 3 [August] 3. THEORETICAL FRAMEWORK The two imensional problem of elasticity involving halfspace soil meia consiere in this stuy is shown in Figure 1, which illustrates general loa f 1(x) an f (x) acting in the positive x an coorinate irections on soil meium lying in the half plane x, The governing equations of plane strain elasticity are the ifferential equations of equilibrium, the stress compatibility equations an the stress-strain laws subject to the bounary conitions. The stress fiels an strain fiels in the half space are require to simultaneously satisfy all the governing equations. The ifferential equations of equilibrium for an elemental part of the soil meium at any arbitrary point (x, y, ) in the Figure 1: Loas f 1(x) an f (x) acting on half space soil semi-infinite mass on the plane are given by: σ xx τ + + b x = (1) τ σ + + b = () When b x =, b =, the ifferential equations of equilibrium are simplifie to be: σ xx τ + = (3) τ σ + = () where b x an b are the two boy forces per unit volume, σ xx an σ are the normal stresses, an τ is the shear stress; an x an are the two imensional Cartesian coorinates of the problem. The compatibility equation in terms of stresses is: σ xx σ τ + = (5) The stress-strain laws for linear elastic homogeneous, isotropic soil for plane strain conitions are given by the three relations: 1 (1 ε xx = ) xx (1 ) E µ σ µ +µ σ (6) 1 (1 ε = ) (1 ) xx E µ σ µ +µ σ (7) τ (1 +µ ) γ = = τ G E (8) where µ is the Poisson s ratio, E is the Young s moulus of elasticity an G is the shear moulus. The bounary conitions are given by: σ xx cos( nx, ) +τ cos( n, ) = f1( x) (9) τ cos( nx, ) +σ cos( n, ) = f( x) (1) where cos (n, x) an cos (n, ) are the irection cosines. On the xy coorinate plane, = an τ = f1 ( x ) = τ ( = ) (11) σ = f ( x ) = σ ( = ) (1). METHODOLOGY: APPLICATION OF THE EXPONENTIAL TRANSFORM METHOD Since, x, we apply the exponential Fourier transformation with respect to x to the governing ifferential equations of equilibrium an the compatibility equations to obtain: σxx τ + e x = (13) 1 F ascicule 3

3 Tome XVI [18] Fascicule 3 [August] τ σ + e x = (1) σ σ xx τ + e x = where λ is the exponential Fourier transform parameter. From Equation (13) Simplifying, σ xx τ e x + e x = (15) (16) λ i σ xxe x + τ e x = (17) Let σ xxe x = σxx ( λ, ) (18) where σ ( λ, ) is the exponential Fourier transform of σ (, ) xx xx τ( λ, ) is the exponential Fourier transform of τ (, ) Then, iλσxx( λ, ) + τ( λ, ) = or τ ( λ, ) = i λσ xx ( λ, ) Also, from Equation (1), an τ e x = τ( λ, ) (19) () (1) τ σ e x + e x = () (3) λ i τ e x + σ e x = iλτ( λ, ) + σ( λ, ) = where σ ( λ, ) is the exponential Fourier transform of σ (, ) an is given by: From Equation (15), we have: Simplifying, Thus, () σ( λ, ) = σe x (5) σ τ σ xx e x e x + e x = (6) xx σ e x ( λ i ) τ e x + ( λ i ) σ e x = (7) σ (, ) (, ) (, ) xx λ + iλ τ λ λ σ λ = (8) A close look at Equation (8) reveals that λ is the exponential Fourier transform parameter, an the only space coorinate variable in the Equation is, hence the equation is an orinary ifferential equation with respect to the space coorinate variable. Thus, Equation (8) is expresse as the orinary ifferential equation (ODE): 15 F ascicule 3

4 Tome XVI [18] Fascicule 3 [August] σ (, ) (, ) (, ) xx λ + iλ τ λ λ σ λ = (9) Using the results from Equations (1) an (), Equation (9) becomes: 1 σ ( λ, ) + σ ( λ, ) λ σ ( λ, ) = λ Alternatively, we have the fourth orer ODE in σ ( λ, ) σ ( λ, ) λ σ ( λ, ) +λ σ ( λ, ) = (31) Applying the exponential Fourier transforms to the bounary conitions at =, From Equation (3), we have: From Equation (33), we have: (3) τ ( x, ) e x = f1( x) e x (3) σ ( x, ) e x = f( x) e x (33) τ ( λ, ) = f ( λ ) (3) 1 σ ( λ, ) = f ( λ ) (35) where f 1 ( λ ) is the exponential Fourier transform of f1(x) an f ( λ ) is the exponential Fourier transform of f (x). 5. RESULTS Solutions for the stresses in the exponential Fourier transform space The vertical stress fiel σ ( λ, ) is foun in the exponential Fourier transform space by solving the fourth orer ODE Equation (31) subject to the bounary conitions. The other stresses σ ( λ, ) an τ ( λ, ) are obtaine from using Equations () an (1). We solve for σ( λ, ) in Equation (31) using the metho of unermine parameters. The nature of the fourth orer ODE suggests the assumption of a trial solution in the form: σ( λ, ) = exp( m) (36) where m is an unetermine parameter. Then for untrivial solutions, the characteristic polynomial becomes: ( m λ ) = (37) The roots are: m = ±λ twice (38) The general solution becomes: λ λ σ( λ, ) = ( c1+ ce ) + ( c3 + ce ) (39) where c 1, c, c 3 an c are the four constants of integration. For boune solutions, the stresses σ are require to ten to ero as, Hence, c 3 =, c = ()-(1) The boune solutions for σ ( λ, ) becomes: (, ) ( c1 ce ) λ xx σ λ = + () The other stresses τ( λ, ), σxx ( λ, ) are obtaine from Equations () an (1). Thus, τ 1 ( λ, ) = (, ) iλ σ λ 1 λ τ( λ, ) = ( c1+ ce ) iλ 1 λ λ τ( λ, ) = c 1 e + c ( e ) iλ (3) () (5) 16 F ascicule 3

5 Tome XVI [18] Fascicule 3 [August] c τ( λ, ) = i ( c1+ c ) e λ λ (6) Also, σ 1 xx (, ) (, ) iλ (7) 1 c λ σxx ( λ, ) = ic ( 1+ c ) e iλ λ (8) c σxx ( λ, ) = ( c1+ c ) e λ λ (9) Using the bounary conitions, Equations (3) an (35), we obtain: (, ) 1 c c τ λ = f λ = i c1 e = i c1 λ λ (5) σ( λ,) = c1( λ ) = f( λ ) (51) c i c1 = f1 ( λ λ ) (5) c f1( λ) f( λ) = = if1( λ) λ i (53) c = f( λ ) + if1( λ) λ (5) c = λ( f( λ ) + if1( λ )) (55) Hence, σ( λ, ) = ( f( λ ) +λ( f( λ ) + if1( λ ))) e λ (56) τ( λ, ) = i f ( λ ) +λ( f ( λ ) + if ( λ)) f ( λ)+ if ( λ ) e λ (57) ( 1 1 ) (, ) i ( f( ) ( 1)( f( ) if1( ))) e λ ( ( ( ) 1( )) ( ( ) ( ( ) 1( ))) (, ) ( ( ) ( )( ( ) ( ))) τ λ = λ + λ λ + λ (58) σ xx = f λ + if λ f λ +λ f λ + if λ e λ (59) σ λ = λ + λ λ + λ (6) xx f f if1 e λ Equations (56), (58) an (6) are the expressions for the solutions for the stresses in the exponential Fourier transform space; an are in terms of the transforme space variables λ an. Solutions for stresses in the physical (problem) omain The stresses σxx (, ), σ (, ), τ (, ) in the problem (physical) omain are obtaine by inversion of the exponential Fourier transforms of the corresponing stresses in Equation (56), (58) an (6). Thus, 1 λ i x σ (, ) = e σ( λ, ) λ (61) where λ λ i x [ 1 ] (6) 1 σ ( x, ) = ( ) ( ( ) ( )) f λ +λ f λ + if λ e e λ iλt iλt 1 1 f ( λ ) = e f () t (63) f ( λ ) = e f () t (6) an t is an integration variable introuce to avoi confusion in the implementation of the integration in the Equation (6). Then, 1 i t i x ( x, ) () e λ f t e λ e λ σ = λ 1 iλt iλt λ λ i x + λ e f() t + i f1() t e e e λ (65) 17 F ascicule 3

6 Tome XVI [18] Fascicule 3 [August] 1 λ 1 () i λ t λ f t e e e i x λ f() t e e i λ t λ σ = λ + λ e i x λ 1 () λ iλt λ i x + i λ f t e e e λ 1 λ +λ i ( t x) i ( ) σ = f( t) (1 +λ) e λ 1 () λ +λ i t x + f t λe λ We note that the integrals are given by: λexp( λ + iλ( t x ) λ) = λexp( λ + iλ( t x)) λ+ λexp( λ + iλ( t x)) λ Similarly, Hence, Similarly, Also, (66) (67) (68) ( ix t ) = ( + ( t x) ) 3 (1 +λ) exp( λ + iλ( t x)) λ= (7) (( x t) + ) (69) ( x t ) f() t + f1() t (71) (( x t) ) σ ( x, ) = + (, ) ( ( ) ( 1)( ( ) 1( ))) λ λ i x (7) λ λ i x λ λ i x ( ) ( 1) ( ) + i ( λ 1) f1( λ) e e λ (73) ( x t) f() t + f1() t τ ( x, ) = ( x t) (( x t) + ) (7) τ x = i f λ + λ f λ + if λ e e λ ( ) τ = i f λ + λ f λ e e λ λ λ i x xx (, ) ( f( ) ( )( f( ) if1( ))) e e (75) λ λ i x λ λ i x ( ) ( ) ( ) + ( λif ) 1( λ) e e λ (76) ( x t) f() t + f1() t σ xx ( x, ) = ( x t) (( x t) + ) (77) σ = λ + λ λ + λ λ ( ) σ xx = f λ + λf λ e e λ Solution for stresses ue to vertical point loa an horiontal point loa at the origin O. Let f 1(x) an f (x) be point loas of magnitues F 1 an F acting at the origin O. Then, f ( x) = Fδ () t (78) 1 1 f( x) = Fδ () t (79) where F 1 an F are constants an δ () t is the Dirac elta function. Then the stresses are ( x t) Fδ () t + F1δ() t (8) (( x t) ) ( x t) F + F1 t (( x t) ) σ ( x, ) = + σ ( x, ) = δ (81) + 18 F ascicule 3

7 Tome XVI [18] Fascicule 3 [August] For F1 =, F where r = x + For F =, F1, ( xf + F1 σ (, ) = ( x + ) (8) ( x t) Fδ () t + F1δ() t ( x t) (83) (( x t) ) ( x t) F + F1 ( x t) t (8) (( x t) + ) τ = + τ = δ x xf + F1 τ = (85) ( x + ) ( x t) Fδ () t + F1δ() t (( x t) ) (86) ( x t) F + F1 (( x t) + ) (87) x xf + F1 xx (, ) (88) σ xx ( x, ) = ( x t) + σ ( x, ) = ( x t) δt xx σ = ( x + ) 3 xf σ xx = (89) r x F τ = r (9) xf σ = r (91) x F σ xx = r (9) xf 1 τ = r (93) 3 F σ 1 = r (9) Solutions for stress fiels ue to uniformly istribute line loa on one sie of the x-axis We consier the specific problem where a uniformly istribute line loa of intensity p acts on one sie of the x axis; in this case, the positive x-axis x. Then, the loa f (x) is expresse as f( x) = ph( x) (95) where H(x) is the Heavisie function given by 1 x H( x) = x < (96) Then, f 1(t) = (97) Then, σ = 1 3 p (( x t) + ) (98) p x t (( x t) + ) (99) (( x t) ) (1) ( ) τ = p ( x t) σ xx = + 19 F ascicule 3

8 Tome XVI [18] Fascicule 3 [August] Let Let Then, tan x t = θ (11) = rsin θ (1) x = rcos θ (13) x t = cot θ= tan θ (1) θ sin θ σ = p 1 + sin θ τ = p (16) θ sin θ σ xx = p 1 (17) 1 1 x where θ= sin = cos (18) r r 6. DISCUSSIONS In this stuy, the exponential Fourier transform metho was use to solve the elastic half plane problem on the coorinate plane involving loas acting on the surface of the bounary where the elastic half plane is assume to be mae of soil meia that is linear elastic, isotropic, homogeneous an semi-infinite in extent with x, an. The governing (fiel) equations are the ifferential equations of equilibrium, the compatibility equation expresse in terms of stress, the kinematic relations an the bounary conitions. The exponential Fourier transformation was applie to the governing ifferential equations of equilibrium an the compatibility equations expresse in terms of stress, to obtain, after algebraic processes, the fourth orer orinary ifferential equation (ODE) Equation (31) expresse in terms of the vertical stress fiels in the exponential Fourier transform space. The exponential Fourier transformation was similarly applie to the bounary conitions to obtain the bounary conitions expresse in the exponential Fourier transform space as Equations (3) an (35). The fourth orer ODE for σ( λ, ), vertical stress in the exponential Fourier transform space, was solve using the metho of unetermine parameters to obtain the general solution given as Equation (39). The requirements for bouneness of the stresses were use to obtain the boune solutions for σ( λ, ) as Equation (). The other stresses τ( λ, ), σxx ( λ, ), in the exponential transform space were obtaine from Equation () using Equations (1) an (), as Equations (6) an (9), respectively. Enforcement of bounary conitions yiele the unknown constants of integration as Equations (51) an (55). Thus the stresses become completely etermine as Equations (56), (58) an (6) in the exponential transform space variables. Inversion of the exponential transforms for the stresses gave the general expressions for stresses in the physical omain space variables as Equations (67), (7) an (77). Particular problems of elastic half plane problems uner vertical an horiontal point loas at the origin, O, were consiere. The stress fiels for elastic half plane problems uner combine vertical an horiontal point loas applie at the origin were foun as Equations (8), (85) an (88). The stress fiels for elastic half plane problems uner vertical point loa applie at the origin were foun as Equations (89-91). The stress fiels for elastic half plane problems uner horiontal point loa applie at the origin were foun as Equations (9-9). Similarly, the specific problem of an elastic half plane uner uniformly istribute line loa p applie to one sie of the x- axis was consiere an the stress fiels were obtaine as Equations (15-17). It was observe that the expressions obtaine were ientical with expressions obtaine by other researchers who use other methos of solution. 7. CONCLUSIONS The conclusions of this stuy are as follows: (i) The exponential Fourier transform metho has been successfully use in this paper to obtain general solutions for the stresses in a linear elastic, isotropic, homogeneous soil meium in the coorinate plane uner bounary loas. (ii) The exponential Fourier transform metho has been successfully implemente in this paper to fin solutions for stresses in elastic half plane meia ue to line loas of constant intensity acting along the x axis. (15) 15 F ascicule 3

9 Tome XVI [18] Fascicule 3 [August] (iii) The exponential Fourier transform metho has been successfully implemente in this work to fin stresses ue to vertical an horiontal point loas applie at the origin O of an elastic half plane meium (on the coorinate plane) consiere linear elastic, isotropic an homogeneous. References [1] Vertical Stress in a Soil Mass. CiVE353/Lectures/stress-loa.pf [] Lecture Note Course Coe BCE Geotechnical Engineering II Stress Distribution in Soil-VSSUT https// lecture18571.pf [3] Frelun D.G., Raharjo H., an Frelun M.D. Unsaturate Soil Mechanics in Engineering Practice. https/books.google.com/books. [] Onah H.N., Osaebe N.N., Ike C.C. an Nwoji C.U. Determination of stresses cause by infinitely long line loas on semi-infinite elastic soils using Fourier transform metho. Nigerian Journal of Technology (NIJOTECH), Volume 35, Number, October 16, pp [5] Ike C.C. Principles of Soil Mechanics. De-Aroit Innovation, Enugu 6. [6] Timoshenko S. Theory of Elasticity. McGraw Hill Book Co. New York 198, pp [7] Das B.M. (8) Avance Soil Mechanics. Thir Eition. Taylor an Francis, Lonon an New York. [8] Verruijt A. (6) Soil Mechanics. Delft University of Technology. [9] Richars R. (1). Principles of Soli Mechanics. CRC Press, Washington D.C. [1] Cummings A.E. (1936) Distribution of stresses uner a founation Transactions, American Society of Civil Engineers, Vol 11, pp [11] Gray H. (1936) Stress Distribution in Elastic Solis. Proceeings of International Conference on Soil Mechanics an Founation Engineering Cambrige Vol, pp [1] Newmark N.M. (19) Stress istribution in soils. Proceeings Purue Conference on Soil Mechanics an its Applications, pp [13] Holl D.L (191) Plane strain istribution of stress in elastic meia. Iowa Engineering Experiment Station, Bull 18 pp.55 [1] Forster C.R. an Fergus S.M. (1951) Stress istribution in a homogeneous soil. Research Report No 1-F. Highway Research Boar, pp.36 ISSN (printe version); ISSN (online); ISSN-L copyright University POLITEHNICA Timisoara, Faculty of Engineering Huneoara, 5, Revolutiei, 33118, Huneoara, ROMANIA F ascicule 3