Synopsis

In a design method presented by GT Houlsby et al (1989), a wheel load is considered to be a long strip load. Assuming plane strain condition and a simple load spread angle concept, the paper proposed a new design method for reinforced unpaved roads. In this paper, instead of a load spread angle concept, Boussinesq's equation of vertical stress for strip loading under plane strain condition is used and the problem of the design of reinforced unpaved roads is modified.

Comparison with other design methods is provided by considering typical design data as given by MR Hausmann.

Introduction: the Houlsby approach

Design methods before Houlsby's method, such as Barenberg (1980), Giroud and Noiray (1981), Raumann (1982) and Sowers et al (1982), mostly concentrated on the membrane effect of the geotextile. Sellmeijer (1982) derived an analysis based on structural membrane theory.

GT Houlsby et al (1989) proposed a new design method for unreinforced and reinforced unpaved roads based on a concept that as vertical loads are applied to a granular layer along with vertical stresses, high horizontal stresses must also develop in that layer. These horizontal stresses are held in equilibrium by shear stresses at the surface of the subgrade and in unreinforced roads these shear stresses have a detrimental effect on the bearing capacity. The resulting horizontal thrust in the soil is partly resisted by horizontal stress in the fill outside the loaded area, but also results in outward shear stresses on the surface of the clay below. The presence of such outward shear stresses reduces the appropriate bearing capacity factor for the clay, possibly to as little as one half of the value for purely vertical loading. If reinforcement is introduced these shear stresses are picked up by the reinforcement, which is put into tension, and purely vertical forces are transmitted to the clay below, allowing the full bearing capacity of the clay to be mobilised.

Modified Houlsby approach

General The modified Houlsby approach (MHA) is based on a similar concept to the Houlsby approach (HA), with the major change being in the method of estimation of vertical stresses within the granular fill due to surface wheel load. Instead of assuming a load distribution factor (tanBETA ) and then computing the vertical stress (s v') at the interface of granular fill and the clay soil subgrade, Boussinesq's equation of vertical stress for a strip load below the centre of a wheel width is used for calculating the vertical stress.

Since the stress below the wheel is more severe than outside the wheel width, only the expression for the centre of the strip load/wheel load is considered for further analysis. The calculated vertical stress is then assumed to be evenly distributed along the effective base width at the interface of the granular fill and the subgrade. This assumption seems to be conservative at first but since the stress field wanders as the wheel wanders, it holds good with an important point presented by Milligan et al (1989), namely, that load can be applied anywhere on the surface of the reinforced fill.

Analysis of stresses within the fill If the average vertical stress due to wheel load, on wheel base of width, 2B, is p, the vertical stress (s zas per Boussinesq, assuming plane strain condition is (Figure 1):

sv' = . z + P(2tan-1(B) + sin(2 tan-1(B)))(1) Where . = unit weight in kN/m 3of the granular fill Therefore, a minimum horizontal force on the plane ad, due to tendency of outward movement of the granular fill of thickness D will be:

D K a . s

v' dz z =0 Therefore from equation 1 DKa . sv' dz =1/2 Ka. D2+2Kap{(Dcot-1(D)+Bln(1 + D2))-B(ln sin tan-1(B))}p. 2 . 2D(2) Where Kais the coefficient of active earth pressure = 1- sinf1+sinf And f is the angle of internal friction of the granular fill material Similarly, assuming that as the fill is pushed outward passive pressures are developed outside the wheel base, the maximum horizontal force on the plane ce, is calculated as 1/2 Kp. D2, where K

pis the coefficient of passive earth pressure, which is equal to (1+ sinf) /(1- sinf).

Allowing also for a frictional force, pBtand, on the base of the tyre, where d is the mobilised footing roughness angle, horizontal equilibrium of the rectangular block abced, results in the following expression (Figure 2):

trB'=1/2(Ka-Kp). D2+2Kap{Dcot-1(D)+Bln(1+D2)-B(ln sintan-1(B))}-pBtand2D(3) Where: t

r= required shear stress at the base of the block, in kPa B' = the effective wheel base width at the interface of granular fill and the soil subgrade, in metres = B + D tan BETA (Figure 1) D = thickness of the granular fill, in metres tan BETA = load distribution factorBETA = load spread angle in degrees (Figure 1) Equation 3 gives the minimum value oft

r, which is required on the base of the block to maintain equilibrium for any given value of p.

Assuming the vertical stress (s

v') at the depth D calculated from equation 1 is uniformly acting on the surface de, (Figure 1):

sv' = pB

+ . D (4) ' Therefore from equation 4, B' and hence, tanBETA can be calculated.

The required bearing capacity factor for the clayey soil subgrade Ncris given by:

Ncr = svr'-svo(5)s uWhere: svr

'= required vertical stress, in kPa (equation 4) su= undrained shear strength of the clayey soil subgrade, in kPa. svo= stress due to overburden soil, in kPa . svo= . D (6) Thus using equations 4 and 6 in equation 5 Ncr = pB (7)uB'The required shear stress factor . ris . r =

tr(8) s

uEquation 3 may be arranged in a nondimensional form, in terms of the required shear stress factor (. r) and the required bearing capacity factor (Ncr

). Thus using equations 7 and 8 in equation 3, the nondimensional form of equation 3 is:

. r=1/2(Ka-Kp) . D2+Ncr{2Ka(Dtan-1(B)+0. 5 ln(1+D2)- ln sin tan-1 (B))- tand}suB p . D B2D(9) This is the equation of a straight line and is shown by the line Go'Ho' in Figure 3. In Figure 3, ABCoCE shows an envelope of available combinat ions of ver t ica l stress, sva

', and shear stress, t

a, on the surface of the clay. This is called an interaction diagram. This is the nondimensional plot of shear stress factor . versus bearing capacity factor Nc(Houlsby et al, 1989).

Design of an unreinforced unpaved road For the assumed depth of a granular fill (Du) the vertical stress (s

v') is calculated using equation 1, which is then equated to equation 4 and the value of effective wheel base width at the interface of granular fill and the soil subgrade (B' ) is obtained. The load distribution factor, tanBETA, can be obtained by using expression (B' - B)/ Du. Using equation 9 and Figure 3, the mobilised/required bearing capacity factor (Ncr

), is obtained (corresponding to point Coin Figure 3). Equation 7 may be now rearranged as follows:

p = Ncr

Su(B' /B) =NcrSu [1+ (Du/B) tan BETA ] (10) Equation 10 gives the vertical stress on the surface of the granular fill due to wheel load, for the assumed thickness of the granular fill (D u). The above steps are repeated for the various assumed thicknesses of the granular fills to get the surface stress due to the wheel load.

The required thickness of the granular fill (D u) for the given wheel load and material properties is then calculated by interpolation or graphically.

Design of a reinforced unpaved roadThickness of the granular fill When the unpaved road is fully reinforced with geosynthetic material at the interface of granular fill and the subgrade soil, the shear stress at the base of the fill is carried by the reinforcement (Figure 4) and the clay below is subjected to purely vertical loading. The clay therefore develops the full bearing capacity factor, N c= 5. 14.

Thus from equation 10:

DrtanBETA = B ( p-1)(11) uWhere: D r= the thickness of granular fill for the reinforced unpaved road.

Equating Equations 1 and 4, and substituting B' = B + D rtanBETA for a depth, D r, the following expression is obtained.

2tan-1( B)+ sin (2 tan-1(B)) = pB (12)rDr B+DrtanBETA The value of D rtanBETA from equation 11 is substituted in equation 12, thus the right hand side of equation 12 becomes known.

For various values of (D r/B), the left hand side of equation 12 can be calculated and plotted as shown in Figure 5. For a known value of the right hand side of equation 12, a corresponding value of D r/B can be obtained from such a plot.

Therefore, for the given wheel base width (2B), the thickness of granular fill (D r) and the load distribution factor (tanBETA ) for the reinforced unpaved road are calculated.

Geotextile tension

The tension induced in the geotextile (T) is obtained from equation 3 (Figure 4) T = t rB'But from equation 8 tr = Therefore, T = . rsuB ' kN/m width (13) Where B' = B + D . r= the required shear stress factor, and is computed from the equation 9 for Ncr= 5. 14 Design procedure For the known surface stress (p) due to wheel load, angle of internal friction (f ) of the granular fill, undrained shear strength (s u) of the subgrade soil, wheel base width (2B), unit weight of the granular material (. ) and the skin friction (d ), the required thickness of the granular fill (D) is computed using the following procedure.

Unreinforced unpaved road For the assumed depth (D), using equation 12, the value of tanBETA and hence the value of B' is calculated. Equation 9 is then used to get the straight line expression for the required stresses. The interaction diagram (Figure 3) is used to obtain the value of mobilised/required bearing capacity factor, Ncr

. Equation 10 is then used to obtain the value of surface stress (p) due to wheel load.

The above steps are repeated by assuming different values of D. The data can then be plotted as calculated (p) versus depth (D) to get the required value of D ufor given surface stress (p) or interpolation can be used.

Reinforced unpaved road The value of DrtanBETA is obtained from equation 11. The value of (D r/B) is obtained by using equation 12 and Figure 5. For the given wheel base (2B) the required value of D rand hence the load distribution factor, tanBETA can be calculated. Tension (T) induced in geotextile is obtained using Ncr=5. 14, from equation 9, and then using equation 13.

Comparison with other design methods Hausmann (1987) reviewed the design procedures in a similar way to Hammit (1970), Giroud and Noiray (1981) and Sellmeijer et al (1982) and presented specific example data.

One set of data he considered for comparison purposes is: Wheel load (P) = 40kN.

Effective tyre print width (dual wheel) (2B) = 0. 49m Effective tyre print length (L) = 0. 24m Number of passes (N) = 1 Unit weight of granular material () = 20kN/m 3Angle of internal friction of the granular fill material (f ) = 35degrees Undrained shear strength (s u) = 20 kN/m2Subgrade friction angle (f s) = 0degrees Skin friction (d ) = 0 Rut depth (r) = 0. 075m Using this data for the HA and MHA, the required thickness of granular fill for the unreinforced and reinforced pavement has been calculated and is presented in Ta b l e 1 .

The table also shows the required thickness using Hammit, Giroud and Noiray and Sellmeijer et al for comparison.

Discussion

Assuming a value of load distribution factor (tanBETA ), the vertical stress (s v') along the plane of symmetry of a wheel base due to unit surface stress has been calculated using equation 4.

Similarly, using equation 1, the vertical stress (s v') at the aggregate-subgrade interface on the axis of symmetry of wheel load is calculated. Both these stresses are plotted in the form of a typical stress profile as shown in Figure 6a. This stress profile clearly indicates that the vertical stress (MHA) is higher than one calculated using simple load spread angle concept (HA). For tanBETA = 0. 354 (the value obtained for the reinforced case by MHA), the stress computed using HA shows that the stress is less and becomes equal to that of MHA at D/B = 1. 6 and then it becomes more compared to MHA (Figure 6a).

A lot of uncertainty is involved in the actual point of application of wheel loads, especially in areas used for loading and unloading, parking and other similar areas. For safe, comfortable, smooth and fast traffic, with low operation and maintenance costs, a rut-free riding surface is desired. To meet these requirements, accurate estimation of stresses is highly essential.

Hence, Boussinesq's equation of vertical stress due to strip loading is expected to be justified, since this approach (MHA) gives high stresses and thus a more conservative thickness requirement of granular fill, as compared to the HA calculation.

In MHA, the value of load distribution factor is found to be less than 0. 5. For reinforced and unreinforced cases, it is not same. In Indian conditions where labourers are unskilled and untrained this seems to be quite reasonable, because hand packing of the sub-base course material may not deliver the same quality as machine method.

Neither the HA nor the MHA consider the membrane effect of the fabric. Hence, indirectly, the designed granular fill thickness gets additional support, as soon as membrane action of the geotextile actuates due to significant rutting.

Conclusion

At first glance the HA and MHA give higher thickness requirements of granular fill for both unreinforced and reinforced unpaved roads, and it may be felt that these methods are uneconomical compared to other techniques. But long term performance must be the criterion rather than any initial cost saving.

This study produces the following noteworthy points a) If the thickness of granular fill is within 2. 878B, MHA must be used instead of HA (Figure 6a).

b) The percentage saving in granular fill thickness due to geotextile is 40. 4% and 29. 5% for HA and MHA respectively, compared to Giroud and Noiray (37. 8%) and Sellmeijer et al (8. 33%).

c) As in HA, anchorage of the edge of the reinforcement is not viewed as important in MHA, but high stiffness may be necessary to realise the potential benefits.

d) The load distribution factor tanBETA in all the methods excluding MHA is either assumed or is determined in the laboratory. Furthermore its value remains the same for an unreinforced and a reinforced unpaved road. But in MHA its value is determined analytically and it is found that it is different for unreinforced and reinforced cases.

e) The mobilised value of the bearing capacity factor N cfor the unreinforced case is assumed to be constant and equal to 3. 14 in Giroud and Noiray's method. However, in HA and MHA this value does not remain constant but varies with the thickness of the granular fill. For the typical data its value is 4. 09 and 3. 91 for the HA and MHA respectively.

Acknowledgements

The work reported in this paper forms part of the postgraduate thesis work of Professor SS Bhosale carried out under the guidance of Professor DR Phatak at the Government Engineering College, Pune, during 1998 and 1999. Facilities provided by the college are gratefully acknowledged.

The library facilities provided by the authorities, particularly Shri RD Kulkarni of CWPRS, Khadakwasala, Pune, are also gratefully acknowledged.

The authors would like to acknowledge the help given in preparing this paper by Professor ST Nigade, lecturer in civil engineering, Government College of Engineering, Pune.

References

Giroud JP and Noiray L (1981). Geotextile reinforced unpaved road design. Journal of the Geotechnical Engineering Division, ASCE, Vol. 107, No. GT9, pp1233-1254.

Giroud JP and Noiray L (1982). Discussion and closure of Geotextile reinforced unpaved road design. Journal of the Geotechnical Engineering Division, ASCE, Vol 108, No GT12, Dec. 1982, pp1654-1670.

Hausmann MR (1987), Geotextile for unpaved roads - A review of design procedures. Geotextile and Geomembranes, Vol 5, No3, Elsevier Applied Science Publishers, England, pp201-223.

Houlsby GT, Milligan GWE, Jewell RA and Burd HJ (1989). A new approach to the design of unpaved roads - Part I, Ground Engineering, Vol22, No 3, pp25-29.

Jewell RA (1996). Soil reinforcement with geotextiles. CIRIA Special Publication 123, Thomas Telford, London, pp235-289.

Milligan GWE, Jewell RA, Houlsby GT and Burd HJ (1989). A new approach to the design of unpaved roads - Part II. Ground Engineering, Vol 22, No 8, pp37-42.

Sellmeijer JB, Kenter CT and Van Den Berg C (1982). Calculation method for a fabric reinforced road, Proc. Second Int. Conf.

on Geotextiles, Las Vegas, USA, pp393-398.

Venkatramaiah C (1993). Geotechnical Engineering. First Edition, Wiley Eastern, New Delhi.

Notation

2BTyre base width at the top of granular fill 2B' Effective tyre base width at top of subgrade B Half tyre base width at top of granular fill B' Effective half tyre base width at top of subgrade D Thickness of the granular fill D

rThickness of the granular fill for the reinforced pavement D

uThickness of the granular fill for unreinforced pavement K

aActive earth pressure coefficient K

pPassive earth pressure coefficient N

cBearing capacity factor due to cohesion for the subgrade Nca

Available bearing capacity factor due to cohesion for the subgrade Ncr

Required bearing capacity factor due to cohesion for the subgrade p Load per unit area s

uUndrained shear strength of the subgrade soil in plane strain T Tension induced in the geotextile ZDepthofinterestfromthegroundsurface. Shear stress factor. aAvailable shear stress factor. rRequired shear stress factorBETA Load spread angle in granular fill tanBETA Load distribution factor. Bulk unit weight of granular filldAngle of friction on base of tyresPressure or stresssv Vertical pressure or stress svrRequired ver t ical stresssvoVertical stress due to overburden soilsv' Vertical stress at a point due to both self weight of soil and surface loadfAngle of friction of granular fill in plane straintShear stresstrRequired shear stresstaAvailable shear stress

## Have your say

You must sign in to make a comment

Please remember that the submission of any material is governed by our Terms and Conditions and by submitting material you confirm your agreement to these Terms and Conditions. Please note comments made online may also be published in the print edition of

New Civil Engineer. Links may be included in your comments but HTML is not permitted.