CHAPTER 3 ANALYSIS OF BEAMS *ON ELASTIC FOUNDATION The continuous beam type of footing system is generally analysed as beam on elastic foundation. Such a structural system for the footing ensures flexural behaviour and minimises the differential settlement. The structural analysis of beams on elastic foundation requires the computation of exact contact pressure distribution satisfying the displacement compatibility between the soil and footing beam. The relative stiffness of the footing and the soil greatly affects the contact pressure distribution and subsequently the distribution of internal forces in the footing beam. The behaviour of soil is very complex as it is rarely homogeneous, generally layered and never perfectly elastic. The deformations are time dependant and partly irreversible. Therefore the numerical methods with good approximations, consistant with the accuracy of the values usually obtained for the soil parameters in routine exploration and testing procedures, provide equally reliable results as the classical rigorous methods for the analysis of beams on elastic foundation. The numerical methods also have the advantage of being easily programmable for the computer. The finite differance method presented in this chapter, 27 i
28 for the analysis of beams on elastic foundation assumes a realistic contact pressure distribution, namely the parabolic distribution, while most of the numerical methods assume a stepped distribution. The computational procedure presented is simple and requires less computational time and effort compared to the finite element methods. The method is also applicable for beams of varying cross section (i.e. moment of inertia ) and varying subgrade modulus. 3.1 LITERATURE REVIEW A number of computational procedures have been developed for the determination of contact pressure distribution and internal forces m the beams on elastic foundation. The classical text on the subject is that of Hetenyi a (1946), who provided the closed form solutions using Winkler's hypothesis, which considered the soil to behave as infinite number of individual independent elastic springs of subgrade modulus as the stiffness. The solutions are based on the following two simplifying assumptions. First, the beam is of uniform cross section, and second the subgrade modulus is constant, irrespective of whether the soil is in compression or tension. Lee and Harrison (1970) augmented the Hetenyi's work by deriving the equations for deflections, moments and shears in the beam on elastic foundation for the case of a moment acting at some point along the beam by superimposing the Hetenyi's general solution for the concentrated load on the beam of finite length and solution for a finite beam subjected to an end moment and concentrated force.
29 Popov (1951) presented the method of successive approximation of contact pressure distribution of which the first approximation being the contact pressure distribution corresponding to infinitely rigid footing. The deflections of the footing were computed by moment area method. Subsequent approximations were made from the first elastic line or from the straight line and the first curve. An iterative method was presented by Baker (1957) for computation of the contact pressure distribution. The method assumed a specific shape of contact pressure distribution to start with and the same was improved satisfying the compatibility of displacements of soil and the beam footing. A numerical method to obtain contact pressure distribution which can deal with soils of homogeneous anisotropy and isotropy and certain cases of stratification was discussed by Barden (1962). The distribution of contact pressure distribution was assumed to consist of steps or blocks of uniform contact pressure. The solution was presented in the form of influence coefficients. Bowles (1974) analysed the beams on elastic foundation using the finite differance method considering the stepped variation of contact pressure distribution. Jagdish and Sharadabai (1977) used the finite differance method to derive the general equations for the combined footing including variable moment of inertia and subgrade modulus. Stepped variation of contact pressure distribution used. Ramaswamy (1977) presented an analytical procedure using finite differance method for the problem of beam resting on compacted cohesive soil taking into account, the nonlinear and time dependant behaviour of soil, by using the nonlinear viscoelastic constitutive equation for
30 soil. It was concluded " The inclusion of nonlinear, time dependant material properties do not significantly affect the contact pressure distribution and moment distribution. The conventional rigid beam solution is an upperbound ". Method of strips was proposed by Dewaiker (1977) to compute the contact pressure distribution below foundation beams. The method consisted of dividing the footing into a number of strips of uniform pressure. Deflections in the beam were computed using beam theory and deflections in the soil medium using Garbonov - Posadov's theory of stresses and displacements in the elastic medium underlain by rigid layer. Equating the deflections of beam and soil, relative to common datum and considering the equillibrium of vertical forces, necessary equations were obtained to solve the unknown contact pressure distribution. The distribution was studied for various depths to rigid layer and for various degrees of relative rigidities of the beam. Finite element method is used extensively for the analysis of beams on elastic foundation by many researchers. Few are cited here. Bowles (1974) formulated the stiffness matrix by combining the conventional beam element with descrete soil springs at the ends of the beam. The degree of accuracy using this element is highly dependant on the number of elements modelled. Miahara and Ergatoudis ( 1976 ) proposed Qa one dimensional line finite elements offering resistance not only to normal forces but to shear and torsional forces. Wang (1983) derived the expressions for the member stiffness matrix and fixed end reactions and moments from the closed form solutions of the governing differential equation for a few cases of transverse loads, as functions of stiffness modulus of both soil and
Y 31 footing. The stiffness matrix and nodal load vectors due to concentrated forces, concentrated moments and linearly varying distributed forces were derived by Ting and Mockry (1984) for the beams on elastic foundation. Patankar (1985) described the application of infinite elements to the strip foundation on elastic continuum for the computation of vertical settlements. A finite element for the analysis of beam - column on elastic foundation using displacement function obtained from the solution of governing differential equation is proposed by Razhaqpur (1989). Sirosh and Ghali (1989) presented a computational procedure for the analysis of reinforced concrete beams on elastic foundation, accounting for reinforcement in concrete by considering the members to have the transformed cross-sectional area of concrete plus the area of steel multiplied by ES -/Ec(t0)f where Es is elastic modulus of steel and Ec(tQ) is the elastic modulus of concrete at age tq. Thus the creep and shrinkage was also accounted for. Alijanabi et. al.(1990) modified the finite element derived by Ting and Mockry to include the effect of shear modulus of the subgrade reactions of the foundation, as well as axial force in the beam. Kurian (1982) remarked, " It is seen from the general solution of the governing differential equation, E If (d4 y / d x4) + ks y = 0 that ^max a X / ks l.e. Ymax a (ks)-3/4, where X = (ks / 4 E If)1/4 and ^ax a 1 /X
32 l. c. Mmax " ^ Since the sensitivity ol the function, depends upon the power in which the variable appears, the above results show that, the moment is much less sensitive to a variation in subgrade modulus than deflection. An important conclusion that follows from this result is the fact that at any rate, the winkler model is more reliable when the criteria for the design is allowable stresses rather than where it is allowable deformation.m Kurian et.al. (1995) nonlinearised the winkler model by defining the subgrade modulus as the function of applied pressure. This was achieved by hyperbolic fitting of load settlement data from a plate load test used to determine the values of the subgrade modulus. The method was illustrated by considering the example of hypar shell footing. The conclusions again indicate that such a model is mseful particularly in a situation where the criteria for the performance is deformation rather than stresses.
33 3.2 COMPUTATIONAL PROCEDURE (Fig.3.1) is, The governing differential equation for the beam (d2 y / d x2) = (Mn / E If) 3.1 The bending moment Mn, at any point n is produced by the loads together with the contact pressure. The contact pressure may be represented by equivalent concentrated reactions, Rn, at the nodal points. Assuming a parabolic distribution of contact pressure, equivalent concentrated reactions (Fig. 3.1) are, Rjl = - (ks h / 24) (7 yi + 6 y2 - ys> 3.2 at the ends and, ^m = (^s h / 24) (7 ym + 6 ym- -i - ym-2) 3.3 Rn = - (ks h / 24) (2 yn--i + 20 yn + 2 yn+i> 3.4 at any intermediate point, where the deflections, y in the upward direction are positive. difference form, Writing the differential equation (3.1) in the finite yn-i - 2 yn + yn+i = (n2 / e if) Mn 3.5 e.g. at node 2, yi - 2 y2 + y3 = (h2 / if) [ (Rj* h) - (PjXh) + (Mx + M2)]
34 fow P2 P3 P f p? i8 r So Si -**1 h k- h----------------------l R> ft ft lit! ----------H *jio LV Rj -(ksh/24) (7)/] +6y2-y3) R2=( ksh/12 ) (y? +ioy2 -f-y3) f1 p p p q=q I D=P LM R, =(h/2) yt ks R2=h^ks f p p ft fio^n R.-(ksh/6) (2y,+y2) ksh/6) (y,+4 y2+y3) Fig.3-1 Contact pressuie distribution
Jb = (- ks h4 / 24 E If) [7 yi + 6 y2 - y3] + (h2 /E If) [(-PjXh) + (Ml + M2)] 3.6 and at node 3, y2-2 y3 + y4 = (h2 / E If) [(R1x2h) + (R2*h) - (P1x2h) - (P^h) + (Mx + M2 + M3)] = (- ks h4 / 24 E If) [16 yx + 32 y2] + (h2 /E If) [(-P1x2h) + (-P2*h) + (Mx + M2 + H3)] 3.7 Similar equations are written upto (m-l)*"*1 node, and the same may be written in the matrix form as, [CY] {y} = - B4 [CM] (y) + (h2 / E If) ({EMP} + {EMM}) 3.8 i.e. ([CY] + B4 [CM]) (y) = (h2 / E If) {EM} 3.9 where, B = (ks h4 / 24 E If)^ [CYM] {y} = (h2 / E If) {EM} 3.10 Thus eqn.(3.10) is a matrix of (m-2) equations. The remaining two equations are obtained from force equillibrium. Hence, m equations to solve for m unknowns. Summing up the vertical forces, (Rf + R2 + R3 +. Rm) - (Pf + P2 + P3 + *Pra) = 0 3.11 i.e, (- ks h / 24) [(7 yx + 6 y2 - y3) + (2y1 + 20y2 + 2y3) +. * >1 = (Pi + P2 + p3 + * * -pm) 3.12
36 Or, in the matrix form - B4 [CM] {y} = (h2 / E If) {EM} 3.13 and summing up the moments of the forces about the last node, [1^(111-1)11 R?(m-2)h - + Rj^jjh] - [PjJm-lJh + P2(m-2)h + + t (Mf + M2 + M2 + * * -Mju) 0 3.14 Or, in the matrix form - B4 [CM] (y) = (h2 / E If) {EM} 3.15 the matrix form as The above equillibrium equations are also written in [CYM} y - (h? / E If) (EH) 3.16 and appended to the equation(3.10). The complete set of equations (3.10) are solved, to obtain the deflections as, {y} = (CYM]"1 {EM} (h2 / E If) 3.17 Or {y} = {y0) (h2 / E If) 3.18 and the reactions, <R) = (~ksh / 24) {CM] (y) 3.19 {R} = -(B4 / h) [CM] {yd} 3.20 The moments are obtained from {M} = (-kg h2 / 24) [CM] (y) + (EM) 3.21 (M) = -B4 [CM] {yq} + (EM) 3.22
37 The above procedure can be conveniently used for beams of varying cross-sections and varying subgrade modulus by imposing appropriate values of B (a non-dimensional parameter combining the subgrade modulus and rigidity of the foundation), at nodal points in all the equations. It is noticed that the moments obtained with only 10 elements are in good agreement with Hetenyi's classical solution. 3.3 NUMERICAL EXAMPLE AND DISCUSSION procedure. A computer program is written for the computational To assess the performance of the present method a simple illustrative example (Fig.3.2) is chosen for which the results are available (Bowles,1974). The comparison is presented in the table 3.1. From the table 3.1, it is evident that author's results agree well with the Hetenyi's classical solution, in comparison with Bowles's finite differance solution cosidering stepped variation of contact pressure distribution for the same number of elements. The effect of rigidity of the footing relative to subgrade modulus on the contact pressure distribution is presented in fig.3.2. lor highly compressible soils, i.e., for soils of low subgrade modulus the footing becomes relatively rigid and hence the contact pressure distribution is almost uniform, whereas for the soils of high subgrade modulus the footing becomes relatively flexible and the much of the load is
(Parametric study) 38
1 1 39 Table 3.1 COMPARISON OF RF.SUI.TS NODE No. 1 & 11 2 & 10 3 & 9 4 & 8 5 & 7 6 DEFLECTIONS (m) AUTHOR 0.0227 0.0213 0.0198 0.0186 0.0177 0.0174 HETENYI 0.023 0.021 0.020 0.019 0.018 0.018 BOWLES 0.0215 0.0204 0.0193 0.0183 0.0176 0.0174 CONTACT PRESSURE (KN/m2) AUTHOR 172.81 162.16 150.74 141.60 134.75 132.47 HETENYI 175.10 159.90 152.76 144.64 137.03 137.03 BOWLES 162.65 154.36 145.83 138.42 133.45 131.71 BENDING MOMENTS (KN m) AUTHOR 0.0 95.77-465.85-856.07-1085.8-1161.9 HETENYI 0.0 95.92-465.2-855.23-1084.6-1160.3 BOWLES 0.25 175.62-325.25-675.18-879.93-948.43
40 shared by soil reactions m a narrow area under the columns instead of uniform distribution. 3.4 CONCLUSIONS 1. The computational procedure described in this chapter for the analysis of beams on elastic foundation assuming a very realistic contact pressure distribution, is simple and easily comprehensible. 2. The computational procedure can also be used for beams of varying cross-sections and soils of varying subgrade modulus by imposing appropriate values of 5 in the set of finite differance eguations.