modeling of beams on reinforced granular beds

8/13/2019 Modeling of Beams on Reinforced Granular Beds

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Modeling of beams on reinforced granular beds

PRITI MAHESHWARI, P. K. BASUDHAR and S. CHANDRA w

Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India.

(Received 11 May 2004; revised 26 November 2004; accepted 10 December 2004)

Abstract. The paper describes a mechanical model for estimating the exural response of astrip footing, supporting a column (imposing a concentrated load), resting on a compactedgranular bed overlying a reinforcement layer for example, geogrids, geomats etc. below whichlies a loose soil deposit. The footing is idealized as a beam and the reinforcing element isassumed to have nite bending stiffness and negligible frictional resistance. The upper andlower soil layers are idealized by a series of linear and discrete springs (Winkler springs) of different stiffness values. To nd the response of such a model the governing differentialequations have been derived and expressed in a nondimensional form. A closed form ana-lytical solution of the same has been obtained subjected to appropriate boundary conditions.Using the present approach the resulting solution for a degenerated case of a long beam isfound to be identical to the same of Hetenyi (1946, Beams on elastic foundations, Universityof Michigan press, Ann Arbor, MI). Parametric studies reveal that the ratio of exural rigidityof upper and lower beam and the ratio of stiffness of the upper and lower soil layers affectsignicantly the response of the foundation.

Key words. geosynthetic, ground improvement, reinforced beds, winkler foundation.

1. Introduction

Modeling of foundations and earth structures reinforced with geosynthetic forpredicting the expected stresses and displacement with in the soil mass is one of theinteresting topics of research in geotechnical engineering. Such modeling is generallybased on continuum mechanics, discrete elements (mechanical modeling) and niteelements. With mechanical modeling, it is possible to simulate the behavior of earthstructures to understand the soilstructure-reinforcement interaction phenomenonand predict the contact pressure distribution and deformation. Idealization of thereinforced earth system by mechanical models is widely used, very often because of its simplicity and familiarity with the geotechnical engineers.

Several lumped parameter models have been developed over the years for soil structure interaction studies (Winkler, 1867; Filonenko-Borodich, 1940; Hetenyi,1946; Pasternak, 1954; Kerr, 1964, 1965; Rhines, 1969). These models have beenfurther extended to analyze reinforced earth problems. Various studies have beenmade to model the reinforced beds and nd their response under external loads.Some of these are Madhav and Poorooshasb (1988), Ghosh (1991), Shukla and

Geotechnical and Geological Engineering (2006) 24: 313324 Springer 2006DOI 10.1007/s10706-004-7548-z

w Corresponding author: Professor S. Chandra, Department of Civil Engineering, Indian Institute of Technology, Kanpur 208016, India. tel: 91 512 259 7029/259 7667; fax: 91 512 2597395; e-mails: [email protected]; [email protected]; [email protected]


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Chandra (1994a, b, c), Yin (1997, 2000). All the above studies idealized the geo-synthetics layer as rough elastic membrane. The only study that has come to theauthors notice which considers the bending stiffness of the geosynthetics layer is dueto Fakher and Jones (2001) who studied the response of a uniformly distributed loadplaced directly on sand bed overlying a super soft clay with the reinforcementembedded in between these two layers. These studies have resulted in improvedmodeling of reinforced beds and better predictions.

2. Statement of the problem

Figure 1 shows a shallow strip footing idealized as an elastic beam (exural rigidityE 1 I 1 ) of length, 2 l , acted upon by a line load Q t imposed by the wall and idealized as

a concentrated load, at the middle of the footing and resting on the surface of acompacted dense soil layer placed over a reinforcing geogrid/geocell/geomat/geo-mattress layer (idealized as beam of exural rigidity E 2 I 2 and same length 2 l (Fig-ure 2) below which lies a natural loose soil deposit. The reinforcing layer is able toresist bending and assumed to be smooth. The compacted dense soil and theunderlying poor soil layers are idealized as weightless Winkler springs of stiffness k 1and k2 , respectively (Figure 2). Thus, it is a 2-dimensional plane strain problem. Theobjective is to nd the exural response of the foundationreinforced bed interactionand perform the parametric studies to nd the relative inuence of various param-eters on the overall behavior of the foundation system.

3. Analysis

Due to symmetry, only one half (x P 0) of the model is considered in the analysis.The analysis of the idealized foundationreinforced bed (Figure 2) is carried out asfollows;

The deection ordinates of the upper (footing) and lower (geosynthetic) beams aredenoted as y1 and y2 , respectively, then the distributed pressure in the foundationunder the upper beam is p1 k1 ( y1 ) y2 ), and under the lower beam, is p2 k2 y2 .The x and y represent the co-ordinates as shown in Figure 2.

Figure 1 . Denition sketch of the problem.

PRITI MAHESHWARI ET AL.314


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The governing differential equations for the upper and lower beams for 0 x l,are as follows (Hetenyi, 1946)

E 1I 1d 4 y1dx 4

p1 k1 y1 y2 1

and

E 2I 2d 4 y2dx 4

p2 p1 k1 k2 y2 k1 y1 2

The pressures p1 and p2 are considered positive when accompanied by positive(downward) deection. From Equation 1, one can write

y2 E 1I 1k1

d 4 y1dx 4

y1 3

Differentiating the above equation, one gets

d 4 y2dx 4

E 1I 1k1

d 8 y1dx 8

d 4 y1dx 4

4

Combining Equations 2 and 4 the following equation can be obtained,

d 8 y1

dx8

k1

E 1I 1E 2I 2E 1I 1

k2

k1E 1I 1 E 2I 2

d 4 y1

dx4

k1k2

E 1I 1E 2I 2 y1 0 5

Equations 3 and 5 are the governing differential equations for the proposed model.R 1 and R 2 are the characteristic lengths of upper and lower beams, respectively, anddened as

R 1 ffiffiffiffiffiffiffiffiffiE 1I 1k14r and R2 ffiffiffiffiffiffiffiffiffiE 2I 2k24r The governing differential equations are nondimensionalized in terms of the fol-lowing nondimensional parameters:

k 1

k 2

Q t

E1I1

E2I2Upper beam

Lower beam

2l

x

y

y1

y2

Figure 2 . Denition sketch of proposed model.

MODELING OF BEAMS 315


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Nondimensional deection of the upper and lower beam, y0

1 y

1E

1I

1=Q

tR 3

1; y0

2 y

2E

2I

2=Q

tR 3

2; respectively (Matlock and Reese, 1960),

Relative exural rigidity of the beams, R E 1I 1=E 2I 2;Relative stiffness of the soil layers, r k1=k2:Nondimensional co-ordinate along the length of the beams, z x/R 1Nondimensional half length of the beam zmax l/R 1Using the above parameters the governing differential equations 3 and 5 can be

written in nondimensional forms as follows

y02 1R

Rr

3=4 d 4 y01dz 4

y01 6and

d 8 y01dz 8

1 Rr R d 4 y01dz 4 Rr y01 0 7

Writing Equation 7 as

d 8 y01dz 8

Ad 4 y01dz 4

By01 0 8

where, A 1 Rr R and B Rr .The roots of the equation 8 arem1;2;3;4 1 i ffiffiffiffiffiffiffiffiffiffiffia b44r and m5;6;7;8 1 i ffiffiffiffiffiffiffiffia b44r

where, a A2 and b ffiffiffiffiffiffiffiffiffiffiffiffiffiA2

4 Bq . Now, introducing the notation,k1 ffiffiffiffiffiffiffia b44q and k2 ffiffiffiffiffiffiffia b44q , the general solution of Equation 8 is, y01 e

k1 z C 1 cos k1z C 2 sin k1z e k1z C 3 cos k1z C 4 sin k1z

ek2z C 5 cos k2z C 6 sin k2z e k2z C 7 cos k2z C 8 sin k2z 9

The Equation (6) can be written as

y02 1R

Rr

3=4ek1z 1 4k41 C 1 cos k1z C 2 sin k1z e k1 z 1 4k41 C 3 cos k1z C 4 sin k1z ek2z 1 4k42 C 5 cos k2z C 6 sin k2z e k2 z 1 4k4

1 C 7 cos k2z C 8 sin k2z

2664

3775

10

The Equations 9 and 10 give the nondimensional deection of the upper and lowerbeam, respectively. The expression for the normalized bending moment of the upperand lower beam can be obtained by differentiating the Equations 9 and 10, which canbe written as

Normalized bending moment of upper beam = d 2 y01

dz2

Normalized bending moment of lower beam = d 2 y02

dz2

Hetenyi (1946) solved the above equations for innite beams and as such couldreduce the constants of integration to four by introducing the boundary condition



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that the deection is zero at the edges of the beams thus getting the constants C 1 , C 2 ,C 5 , and C 6 equal to zero. However, in the present case the beams are of nite lengthsand, as such, it is necessary to take appropriate boundary conditions to solve theabove equations. These boundary conditions for the present problem are as follows:

For the upper beam, at the point of application of load, i.e., at x 0, slope of thedeected shape of beam is zero and shear force is Q t /2. At the edge of upper beam,i.e., at x l, the bending moment and shear force are zero as beam end is a free end.For the lower beam, which is within the foundation soil, at point x 0, slope of deected shape of the beam and shear force are zero and at x l, bending momentand shear force are zero. From the above-mentioned boundary conditions one getsthe following nondimensional equations.

For upper beam

At z 0; dy01

dz 0 and

d 3 y01dz3

12

At z zmax ; d 2 y01

dz2 0 and

d 3 y01dz3

0

For lower beam

At z 0; dy02

dz 0 and

d 3 y02dz3

0

At z zmax ; d 2 y02

dz2 0 and d 3 y02

dz3 0

11

From the above eight equations, eight linear equations are obtained and solved byCholesky Decomposition Scheme to get the unknown constants C 1 to C 8 . Using theseconstantsin the Equations 9 and 10 the valuesof deections can be computed and theseequations canbe differentiated to getappropriate expressions forcomputing the valuesof bending moment and shear force of the upper and lower beams along their length.

As such, the present solution is more general and Hetenyis solution (1946) be-comes a particular case of the present one. Another aspect of the present analysis,which can be contrasted with that of Hetenyi, lies in the fact that in the present studythe reinforcing beam is a physical entity and in Hetenyis analysis the beam wasintroduced to remove the deciency of the Winkler model which lacks continuity.

4. Results and discussions

As the problem considered is symmetric, only half of the beam (0 z zmax ) isanalyzed. To validate and check the results from the present analysis, the results areobtained for the particular case when both the beams are long enough to beconsidered as innite and compared with the results given by Hetenyi (1946) for



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innite beams and are found to be identical. However for the sake of completeness

the comparison is presented in Figure 3.Parametric studies are carried out to study the effect of various parameters on the

behavior of the foundation. The ranges of various parameters are worked out bytaking practical range of parameters and are given below:

Coefficient of subgrade reaction for dense sand 125375 MN/m 3 (Das, 1999)

Nondimensional length of the beams ( zmax ) 220.

Relative exural rigidity of the beams ( R E 1 I 1 /E 2 I 2 ) 150.Relative stiffness of the soil layers ( r k1 /k2 ) 120.

Figure 4 shows the variation of normalized deection of the upper beam along theco-ordinate direction for different beam lengths ranging from 2 to 20 for particularvalues of R and r , 10 and 5, respectively. It is seen that the maximum deection, of the upper beam occurring at the center of the beam, does not vary at all for all beamshaving a normalized length greater than 6; all beams having normalized lengthgreater that 16 show almost identical deection behavior. Thus, it can be concludedthat for zmax 16, the beam would behave as an innite beam. For beam length lessthat 16 the predicted deection at the end is not zero as expected for innite beamsbut the difference between the values are marginal. For the parameters considered inthe study, heaves of magnitude of 0.19 and 0.14 are observed at the edge of upper

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 2 4 6 8 10 12 14 16 18

Normalized Distance from Centre (z)

N o r m a

l i z e

d

D e

f l e c

t i o n o

f t h e

b e a m

Present StudyHetenyi Solution (1946)

Deflection of Upper Beam

Deflection of Lower Beam

r = 5, R =10

Figure 3 . Comparison of deection of beams with Hetenyi (1946).



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beam for zmax equals to 4 and 8, respectively. For zmax greater than 10, rst thenormalized deection of upper beam reduces from its maximum value at z 0, thenheave is observed from z equals to 5.511 and then again deection becomes positive

which becomes almost zero when z is greater than 16. This implies that upper beamgets lifted up from the ground surface from z 5.5 to 11 for zmax greater than 10.Similar observation is made from Figure 5, with respect to the predicted deectionsfor the lower reinforcing beam, only difference being that the magnitude of deectionof the lower beam is lesser than that of the upper beam. As soil can not take tension,the model can be further improved to take care of the same.

Effect of relative exural rigidity of beams designated by the ratio R has beenstudied for the ratio r 5 and zmax 8. It is observed from Figure 6 that as theratio R varies from 5 to 50, the deection of the beam increases although thisincrease is not signicant. When the ratio R is equal to 1, the predicted value of maximum deection is lower than those obtained for other values of ratio R. Fig-ure 7 shows the corresponding variation for the lower beam. The maximum nor-malized deection of lower beam at the center, i.e., at z 0, reduces by 55% as ratioR increases from 1 to 50. Around z 5.3, the deection of beam for all consideredvalues of ratio R, is almost same and at the edge of the lower beam the normalizeddeection reduces by approximately 77% for the same increase in ratio R. Thisprobably can be explained as follows; as the ratio R increases, the exural rigidity of upper beam is more as compared to that of lower beam, which distribute the load tothe lower beam in such a way that it is subjected to lesser stress level causing lesserdeection for higher values of ratio R.

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10 12 14 16 18


N o r m a

l i z e

d D e

f l e c

t i o n o

f U p p e r

B e a m

( y 1

' )

R = 10, r = 5

z max =2

4

8 10 1216

Figure 4 . Effect of length of the beams on deection of upper beam.



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-0.05

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10 12 14 16


N o r m a

l i z e

d

D e

f l e c

t i o n o

f L o w e r

B e a m

( y 2

' )

R = 10, r = 5

4

8 12 1610

14

zmax =2

Figure 5 . Effect of length of the beams on deection of lower beam.

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5 6 7 8

Normalized Distance from Center (z)

N o r m a

l i z e

d D e

f l e c t

i o n o

f U p p e r

B e a m

( y 1

' )

r = 5, z max =8

R=1

5

1020

50

Figure 6 . Effect of relative exural rigidity on deection of upper beam.



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-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8


N o r m a

l i z e

d D e

f l e c

t i o n o

f L o w e r

B e a m

( y 2

' )r = 5, z max =8

R = 1

5

10

20

50

Figure 7 . Effect of relative exural rigidity on deection of lower beam.

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5 6 7 8


N o r m a

l i z e

d D e

f l e c t

i o n o

f U p p e r

B e a m

( y 1

' ) R = 10, z max =8

r = 20

10

5

21

Figure 8 . Effect of relative stiffness ratio on deection of upper beam.



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The effect of variation of stiffness ratio of the soil layers, r, on the normalizeddeection of upper beam is shown in Figure 8 for ratio R

10 and zmax

8. At the

center, i.e., at z 0, the deection decreases by 87% as ratio r is reduced from 20 to1. For ratio r equal to 1, the deection rst decreases and heave is observed and atthe edge of the beam the deection is negligible but when the ratio r varies from 2 to20, heave is observed beyond z P 6 and the corresponding maximum value of heaveincreases from 0.034 to 1.73 the relative difference between these being very high.Higher values of ratio, r , signies the presence of a weaker soil below the strong topsoil and as such the corresponding deections are greater. Similar behavior isobserved for lower reinforcing beam.

Figures 9 and 10 show the variation of normalized bending moment, of upper andlower beams, respectively, along the length of the beams for the ratio R 10, and

zmax 8. The gures also bring into focus the effect of the variation of the ratio r onthe overall behavior. From Figure 9, it is observed that for the upper beam maxi-mum positive (sagging) bending moment occurs at z 0, which decreases byapproximately 43% as ratio, r, decreases from 20 to 1. The section, at which max-imum negative (hogging) bending moment occurs shifts towards the edge of thebeam as the ratio, r, increases. Similar kind of behavior has been observed for thelower reinforcing beam also (Figure 10). The maximum positive normalized bendingmoment for the lower beam decreases from 0.043 to 0.023 as ratio r decreases from20 to 1.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5 6 7


N o r m a

l i z e

d B e n

d i n g

M o m e n

t f o r

U p p e r

B e a m

r=1

2

5

10

20

R = 10, z max = 8

8

Figure 9 . Effect of variation of relative stiffness of soil on bending moment of upper beam.



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5. Conclusions

The following generalized conclusions can be drawn from the study reported above:(1) For innite beams the solutions obtained from the present study and those of

Hetenyis model (1946) are identical and thus validate the correctness of thesolution procedure.

(2) Beyond the normalized length of beams equal to 16, there is practically nochange in the normalized deection of beams and the edge deection beingnegligible, the beams can be considered to be of innite length for the range of parameters considered.

(3) Relative exural rigidity of beams represented by ratio, R , does not affect muchthe normalized deection of upper beam (strip footing) but affects the normal-ized deection of lower beam (the geosynthetic reinforcement) signicantly. Thedeection of lower beam can be reduced to the extent of 77% by increasing theratio, R, from 1 to 50.

(4) The relative stiffness of soils, i.e., ratio, r, has signicant inuence on the nor-malized deection of upper beam as well as that on lower beam. At the center of beams, the reduction in normalized deection of beam is 87% and 23% forupper and lower beam, respectively, for corresponding decrease in the ratio, r,from 20 to 1. For the upper beam heave is observed at the edge.

(5) Maximum normalized positive bending moment occurs at the center for upperbeam, while at the edge, it is zero. The maximum positive normalized bending

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 1 2 3 4 5 6 7


N o r m a

l i z e d B e n

d i n g

M o m e n

t f o r

L o w e r

B e a m

r=1

2

5

1020

R = 10, z max = 8

8

Figure 10 . Effect of variation of relative stiffness of soil on bending moment of lower beam.



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moment can decrease to the extent of 43% for decrease in ratio, r, from 20 to 1.The position of section of maximum negative normalized bending moment shiftstoward the edge of the beam as ratio, r , increases. Similar kind of behavior hasbeen observed for lower reinforcing beam.

References

Das, B.M. (1999) Principles of foundation engineering , 4th Edition, PWS Publishing, USA.Fakher, A. and Jones, C.J.F.P. (2001) When the bending stiffness of geosynthetic reinforce-

ment is important, Geosynthetics International , 8 (5), 445460.Filonenko-Borodich, M.M. (1940) Some approximate theories of the elastic foundation, Uch.

Zap. Mosk. Gos. Univ. Mekh ., 46, 318 (in Russian).Ghosh, C. (1991) Modelling and analysis of reinforced foundation beds , Ph.D. Thesis,

Department of Civil Engineering, Indian Institute of Technology, Kanpur, India.Hetenyi, M. (1946) Beams on elastic foundations , University of Michigan Press, Ann Arbor,MI.

Kerr, A.D. (1964) Elastic and viscoelastic foundation models, Journal of Applied MechanicsDivision, ASME, 31, 491498.

Kerr, A.D. (1965) A study of a new foundation model, Acta Mechanica , 1, 135147.Madhav, M.R. and Poorooshasb, H.B. (1988) A new model for geosynthetic-reinforced soil,

Computers and Geotechnics , 6, 277290.Matlock, H. And Reese, L.C. (1960) Generalized solutions for laterally loaded piles, Journal of

the Soil Mechanics and Foundations Division, ASCE , 86 (5), 6391.Pasternak, P.L. (1954) On a new method of analysis of an elastic foundation by means of two

foundation constants, Gosudarstvennoe Izdatelstro Liberaturi po Stroitelstvui Arkhitekture ,Moscow (in Russian).

Rhines, W.J. (1969) Elastic-plastic foundation model for punch shear failure. Journal of Soil Mechanics and Foundations Division, ASCE , 95 (3), 819828.

Shukla, S.K. and Chandra, S. (1994a) The effect of prestressing on the settlement character-istics of geosynthetic-reinforced soil, Geotextiles and Geomembranes , 13, 531543.

Shukla, S.K. and Chandra, S. (1994b) A study of settlement response of a geosynthetic-reinforced compressible granular ll-soft soil system, Geotextiles and Geomembranes , 13,627639.

Shukla, S.K. and Chandra, S. (1994c) A generalized mechanical model for geosynthetic-reinforced foundation soil, Geotextiles and Geomembranes , 13, 813825.

Winkler, E. (1867) In Die Lehre von der Elastizitat und Festigkeit, Domonicus, Prague, p. 182.Yin, J.H. (1997) Modelling geosynthetic-reinforced granular ll over soft soil, Geosynthetic

International , 4 (2), 165185.Yin, J.H. (2000) Comparative modeling study on reinforced beam on elastic foundation,

Journal of Geotechnical and Geoenvironmental Engineering Division, ASCE, 26 (3),265271.


modeling of beams on reinforced granular beds

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