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The Winkler modeland i t s application to soil
TheoreticalInvestigation
An approximate general solution1 has been presented forcontact pressures of finite beams estingon idealizedmodels of the soil. Theoundation models treatedinclude:-
(a) Winkler model of isolated springs;(b) homogeneous anisotropic half-space, isotropy being
(c) non-homogeneous half-space, compressibility de-
The solution is based on equating the beam and soil
deflexions. Inspection of the general form of theseequations showed that as the relative flexibility of thebeam increased so the beam deflexion term began tooutweigh the soil deflexion term. This suggested that thesolution fora flexible beam would be nearly independent
of the complexity of the foundation model.Hetenyiz, Biot3 and Vesic4 indicated that for infinitely
long flexible beams the homogeneous isotropicelastichalf-space solutionpproaches the simpler Winklersolution. This is apparently a special case of the moregeneral trend outlined above.
I t was decided, therefore, to take the solution for thehomogeneous isotropic model, compare it with thesolution for the simplest foundation model, the Winkler,and o investigate a) how closely the two olutionsapproach each other; (b) the effect of relative flexibility;and (c) for similar solutions, the form of the relationshipbetween the respective moduli E , and k.
The approximategeneral solution1 was used to plot the
isotropic half-space distributions. The Winkler distribu-tions were obtained by using a Deuce electronic digitalcomputer o produce a veryaccurateset of influencecoefficients from the rigorous solution2 to the Winkler
equation E I- ky =W@).dx4
a special case;
creasing with depth.
d4Y
The isotropic half-space solution is governed by the
dimensionless parameter 1040 =- and he
Winkler solution by 0 = -Jrn.E I
Let 1040 =NO4 or N =1040/ 8 4 .
Distributionsor ~zr covering the ractical range0<o <1 -0 ere plotted and the value of 8 giving theclosest possible Winkler distributionobtained n eachcase from the computed influence coefficients.
4 E I ' l-v2
I 104 0 1 100 ~200~300~400(500~1000~5000~0,000 II 1 - 1 - l - l - 1 - l - 1 - I 1orresponding 8 1.5 2 .3 2.7 3 .03 .2 3.8 5.7 .75
It was observed that for 82 2 . 7 5 the two solutionswere almostdenticalith N constant at 4 - 8 .
7~L4 E , bkL4 E
4 E I 1-19 4 E I l-v2
Thus- - 4 . 8 -andbk =0 . 6 5 2
Working from the rigorous solution, possible forinfinitely long beams on an isotropic half-space, Vesic4
*LAING BARDEN MSc PhD AMICE
obtained a similar relationship bk =0 65
12
Jg -E,- Forpractical cases 22/Egannotdepart1-02
much from unity, so there is general agreement.Vesic's limit was also very close to 8 =2 -75 .
Beams are usually lassed as stiff O < 8<3, intermediate3< 8<7 and flexible 7< 8<10 . The act that helimit 8 =2 . 7 5 in the case considered is so low in thepractical range suggests that regardless of the complexityof the foundation model, similarity of solution willexist for all but stiff beams.
ExperimentalInvestigation
The stress-deformation behaviour of sand is so complexthat no idealized elastic foundation model is likely toprove effective. From the above considerations it wouldappear, however, th at regardless of the complexity of itsbehaviour, contact pressure distributions will be closelygiven by the Winkler solution for all but stiff beams onsand.
The following laboratory scale experiments weredesigned to investigate this hypothesis.
Contact pressure distributions are usually investigated
byhe double differentiation of bending momentscalculated from strain gauge readings, but this processlacks precision. I t was considered preferable to measurethe contactpressures directly by means of cells set in thebeam-sand interface.
Pressure Cells
Taylor has shown that to prevent arching and under-registration of pressures the ratio of the diameter to thedeflexion of the diaphragm of a cell should exceed 1000.Based on the vibrating wire principle a stable cell wasdeveloped which had a relevant ratio of 2000 and whichwas accurate to 0 - 5 lb/in2 in the 0- 0 lb/in2 pressurerange.
Beam
The basic beam was 36 x 4 x 4 in. mild steel. Elevencells were fitted at intervals along the beam centre-linewith their diaphragms flush with the beam face. Theywere secured only in the region of the neutralaxis so thatflexure of the beam did ot affect the iaphragmdeflexion. This basic beam was stiffened by channelsections for tests at higher E I values.
The load was applied through a hinged knife-edge byscrewing against a loading frame, and was measured by a5ton proving ring.
Sand Foundation
I t was essential to obtain very even compaction of thesand foundation as any tendency to hard or soft spotscaused distortion of thecontact pressure distribution.A dense, homogeneous foundation was obtained by con-taining dry sand passing a No. 25 sieve in a timber box
60 X 18 x 15 in.deep whichwas vibrated aterallywith shutter vibrators. The top surface was struck offflush to receive the beam.
*Lecturer in Engineering, University of Manchester.
THE STRUCTURALNGINEEREPTEMBER 1963 No 9 VOLUME 41 21
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Results
As anticipated rom he esults of Fabers he ateraldistribution is parabolic and it was necessary to reducethe measured centre-line pressures by a factor of about0 -7o make them balance the vertical applied loads. Thecorrected longitudinal distributions are plotted in Figs1 to 5 as the full line.
On dividing the pressure diagram by he deflexiondiagramhe ariation of the modulus of subgradereaction k was obtained. Although k was constant along
thebeam only orcentralpoint oading compare asimilar result on a silt foundation-Vesic4) an average
value of k could be assessed and used to calculate theparameter 8.
The corresponding theoretical Winkler solutions wereobtained from the computed tablesf influence coefficientsand have been plotted in Figs 1 to 5 as the broken lines.
Discussion
( l ) Unequal ompaction anddisturbanceduring hefinal striking-off of the surface of the sand foundationoften esulted in local hardand soft spotsand heexperiments were usually repeated a number of timesbefore a consistent set of pressure cell readings, exhibiting
acceptable scatter, were obtained.(2) k , as calculated, was generallynot onstant, itsvariation along the beam being affected by the pa tternof applied oading, confirming the over-simplificationof the Winkler model.(3) SchultzeG has shown that under rigid foundationsin sand at a considerable depth edge stresses do occur,the surchargeproviding the necessary shear strengthto allow their development. This indicates that sand canresemble a continuum.(4 ) Theesults of Figs 1 to 5 confirmed that for8>2 -75 the Winkler solution gives a close approxima-tion of contact pressure- distribution on dense sand.
Consideration of 2 and 3 above suggests that hefindings of 4 are a consequence of the hypothesis underinvestigation rather han he intrinsicvalue of theWinkler model.
The theoretical deductions apply generally to almostany soil but have been verified experimentally only fordense sand. I t is probable, however, that the traditionalWinkler solution is satisfactory engineering approxima-tion or llbut stiff beams. In practice the maindifficulty is in assessing a value for k , the hypotheticalmodulus of subgrade reaction-see Terzaghi7.
References
1.
2.
3.
4.
5 .
6.
7.
R a r d y , L., 'Distribution of contact pressure under founclations , Gkotechnique, Vol. 12, No. 3, September 1962.Hetenyi,M., B e a m s on Elastic Foundat ion,Univ. Michigan Press,
1946.Biot, M . A., 'Bending of an infinite beam on a n elastic founda-tion ', Trans. ASME, Vol. 59, 1937.Vesic, A. B., 'Beams on elasticsubgrade and he Winklerhypothesis 1, Proc . 5th I n t . Conf. Soil Mech., Paris, 1961Faber, 0.. Pressure distributions under bases ', Struct. En g . ,Vol. XI, March 1933.Schultze, E., 'Distribution of a stress beneath a rigid founda-tion ', P r o c . 5 t h I n t . Conf. Soil M e c h . , Paris, 1961.Terzaghi, K. ,'Evaluation of coefficientsof subgrade reaction ',Gkotechl.tique,Vol. 5 , No. 4, December 1955.
5 0 ' -
F ig l
10.5 TON I 0.5 TON
F i g 2
F ig 5
280 VOLUME 41 No 9 SEPTEMBER 1963 TH ETRUCTURAL ENGINEER
