Razo Uki
Proceedings of the Institution of Civil Engineers Geotechnical Engineering 163 April 2010 Issue GE2 Pages 101–106 doi: 1
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Proceedings of the Institution of Civil Engineers Geotechnical Engineering 163 April 2010 Issue GE2 Pages 101–106 doi: 10.1680/geng.2010.163.2.101 Paper 800012 Received 12/02/2008 Accepted 13/07/2009
Sabah Said Razouki Professor of Civil Engineering, Nahrain University, Al-Jadiriya, Iraq
Dahlia A. Al-Zubaidy Formerly postgraduate student in Civil Engineering, Nahrain University, Al-Jadiriya, Iraq
Keywords: geotechnical engineering/rock mechanics
Elastic settlement of square footings on a two-layer deposit S. S. Razouki
PhD, Dipl.Ing, Ceng, MASE
and D. A. Al-Zubaidy
Two charts for elastic (immediate) settlement at the centerline of a flexible square footing on Burmister twolayer elastic system are presented in this paper. The charts are developed for different values of the modular ratio E1 /E2 [where E1 refers to the modulus of elasticity of the first (top) layer and E2 corresponds to the second (bottom) layer] and relative layer thickness h/B (where h refers to the first layer’s thickness and B to the side length of square footing ). The first chart is devoted to the case of a soft top layer on a stiffer bottom layer, while the second chart is for a stiff top layer over a softer bottom layer. The analytical solution for surface deflection was based on Burmister analysis and extended to the case of square footings. The paper reveals that the surface settlement at centerline decreases with increasing modular ratio for E1 /E2 > 1 and it increases with decreasing modular ratio for E1 /E2 < 1 especially for the higher values of h/B.
NOTATION a radius of uniformly loaded circular area B width (least dimension) of rectangular footing; diameter of circular footing (B ¼ 2a); side length of square footing Df embedment depth E modulus of elasticity (Young’s modulus) of soil E1 modulus of elasticity of first (top) layer E2 modulus of elasticity of second (bottom) layer E1 /E2 modular ratio F0 surface deflection factor for flexible circular footing on Burmister two-layer elastic system FS surface deflection factor for flexible square footing on Burmister two-layer elastic system, FS ¼ F0 /ˇð f (xi ) the function at the ith point G function of N and m h first layer’s thickness for two-layer elastic system I5 influence factor for flexible rectangular footing for different values of L/B J0 Bessel function of the first kind, of order zero J1 Bessel function of the first kind, of order one L largest dimension (length) of rectangular footing m a parameter N ratio used in Sections 2, 3 and 4 ¼ (E1 /E2 1)/ (E1 /E2 + 1) q intensity of contact pressure in units of E Geotechnical Engineering 163 Issue GE2
r Se wi w0
xi äe í í1 í2
MSc radial distance (horizontal distance) from centre of loaded area to computational point immediate settlement (elastic settlement) weight of function at ith point vertical surface deflection at any point on the surface at horizontal radial distance r from centre of uniform loading position of function at ith point surface elastic settlement at centreline of a flexible square footing Poisson’s ratio Poisson’s ratio for first (top) layer Poisson’s ratio for second (bottom) layer
1. INTRODUCTION In general, the total settlement of a foundation consists of three parts: immediate settlement, primary consolidation settlement, and secondary consolidation settlement (Das, 1985, 2004). The immediate settlement is also referred to as the elastic settlement (Das, 1985) or initial settlement (Kezdi, 1964). According to Bowles (1988, 1996), the immediate settlement takes place as the load is applied, or within a time period of 7 days. According to Das (1985) and Bowles (1988), the immediate settlement is the predominant part of the settlement in granular soils. In practice, the immediate settlement is usually calculated from the theory of elasticity using the linearly elastic homogeneous and isotropic Boussinesq halfspace (Das, 1985). This means that the elastic soil layer extends to an infinite depth. This assumption is far from being true in many cases in practice. However, for an elastic soil layer underlain by a rigid incompressible base, Das (1985) suggested an approximate solution for the determination of immediate settlement on the basis of a Boussinesq solution, and Poulos (1967) and Poulos and Davis (1974) developed charts that enable the elastic settlement for an elastic layer underlain by a rough rigid base to be determined for various cases of loadings. Similarly, Razouki and Issa (2001) presented charts for determining the elastic settlement due to flexible embankment loading on an elastic layer underlain by a rough rigid base. In practice, the foundation engineer faces the problem of layered soil systems. The simplest case is that of a two-layer soil system with a modular ratio greater or less than 1, for which charts of immediate settlement are required. This paper
Elastic settlement of square footings on a two-layer deposit
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101
is devoted to such a problem, but for the special case of square footings only.
factor for a flexible circular footing on a Burmister two-layer elastic system.
2. SURFACE SETTLEMENT DUE TO UNIFORMLY LOADED RECTANGULAR AREA ON BOUSSINESQ HALF-SPACE The elastic surface settlement at the centre of a uniformly loaded flexible rectangular area can be found using the following equation based on Boussinesq half-space (Das, 1985).
The surface deflection factor for the case of í1 ¼ í2 ¼ 0.5 for a circular footing is given by (Huang, 1969) F0 ¼
4
ð1
f ð mÞdm
0
where 1
qB ð1 í 2 Þ I 5 Se ¼ E
5
where Se is the surface elastic settlement at the centreline of a flexible rectangular footing; q is the intensity of contact pressure; B is the width of the rectangular footing; E is the Young’s modulus of elasticity of the soil; í is the Poisson’s ratio of the soil; I5 is an influence factor (depending on L/B for surface deflection); and L is the length of the rectangular footing. For a square footing, I5 ¼ 1.122 (according to Das, 1985), and Equation 1 yields
2
qB ð1 í 2 Þ äe ¼ 1:122 E
where äe is the surface elastic settlement at the centreline of a flexible square footing. 3. SURFACE DEFLECTION OF BURMISTER TWOLAYER ELASTIC SYSTEM For the purpose of pavement design for highways (Huang, 1993; Yoder and Witczak, 1975) and airports (Horonjeff and McKelvey, 1994; Huang, 1993), Burmister (1943) solved the problem of a two-layer soil system representing a stiff soil layer with a modulus of elasticity E1 on top of a softer layer with a modulus of elasticity E2 subjected to flexible uniform circular loading. Each layer of the system is assumed to extend to infinity in the lateral direction. The top or first layer has a finite thickness, whereas the bottom or second layer extends to infinity in depth. Each layer is assumed to be linearly elastic homogeneous and isotropic, with no slippage at the interface between the two layers. Burmister (1943) assumed that the top and bottom layers have the same Poisson’s ratio of 0.5. According to Huang (1969), the surface deflection at the centreline of uniform flexible circular loading on top of a twolayer soil system is given by
3
w0 ¼
qa F0 E2
Geotechnical Engineering 163 Issue GE2
mr ma E1 J1 G m h h E2
E1 G m E2 6 ¼
1:5ð1 þ 4Nme2 m N 2 e4 m Þ ð E 1 =E 2 Þ 1 2N ð1 þ 2m2 Þe2 m þ N 2 e4 m m
N¼
7
E 1 E 2 E 1 =E 2 1 ¼ E 1 þ E 2 E 1 =E 2 þ 1
m is a parameter; E1 /E2 is the modular ratio, where E1 refers to the modulus of elasticity of the first (top) layer and E2 refers to the modulus of elasticity of the second (bottom) layer; J0 is a Bessel function of the first kind, of order zero (Watson, 1966), given by J0
8
mr h
¼
1 X ð1Þ K ð mr=2hÞ2 K K¼0
ð K!Þ2
and J1 is a Bessel function of the first kind, of order one (Watson, 1966), given by 9
J1
pffiffiffi2 Kþ1 X 1 ð1Þ K mB=2h ð mB pffiffiffi ¼ K!ð K þ 1Þ! h ð K0
4. SURFACE SETTLEMENT OF SQUARE FOOTING ON BURMISTER TWO-LAYER ELASTIC SYSTEM To obtain the surface elastic settlement of a flexible square footing with uniform loading on a Burmister two-layer soil system, it is satisfactory for all practical purposes to replace the square footing by an equivalent circular footing having the same area and the same centre as the square footing, as follows
10
where w0 is the vertical surface deflection at any point on the surface at a horizontal radial distance r from the centre of the circular loading; q is uniformly distributed load; a is the radius of the circular loading area; E2 is the modulus of elasticity of the second (bottom) layer; and F0 is the surface deflection 102
f ð mÞ ¼ J 0
B ða2 ¼ B2 or a ¼ pffiffiffi ð
where a is the radius of the equivalent circular footing and B is the width or side length of the square footing. Thus the elastic surface settlement at the centreline of a square footing on a Burmister two-layer system can be obtained as
Elastic settlement of square footings on a two-layer deposit
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Accordingly, the integrand f (m) was assigned to a rather wide range of m, beginning from zero, until the effect of the upper limit on the integral became insignificant. First, the pffiffiffi zeros of the Bessel function J 1 (mB=h ð) were computed by using the computer program SDF (surface deflection factor) written in Mathcad (2001) and checked by comparing them with those tabulated by Watson (1966), and excellent agreement was obtained. The area between every two consecutive zeros was calculated using the Gaussian quadrature procedure. The Gaussian quadrature was selected to compute these areas owing to its highest possible degree of precision (Engel, 1980).
qa F0 E2
w0 ¼
qB pffiffiffi F0 E2 ð
¼
or
w0 ¼
11
qB FS E2
where w0 is the vertical surface deflection at any point on the surface of the square footing, and Fs is the surface deflection factor at the centreline for the case of a square footing on Burmister two-layer elastic system, given by
The Gaussian 15-point formula is (Selvadurai, 1979) ð1
f ð x Þdx ¼
16 1
F0 Fs ¼ pffiffiffi ð
12
Making use of Equations 4 and 12, the surface deflection factor Fs becomes 1 Fs ¼ pffiffiffi ð 13
ð1 f ð mÞdm 0
ð 1 1 mr mB E1 J 1 pffiffiffi G m dm J0 ¼ pffiffiffi h E2 ð 0 h ð
It is obvious from this equation that the integrand f (m) is very complicated, and therefore it is very difficult or rather impossible to solve the integral analytically. Thus numerical methods should be used for computing the deflection factors. For the case of the centreline 14
J0
mr h
1 Fs ¼ pffiffiffi ð
ð1 J1 0
When the limits of the integral are a and b, the integration is computed using the Gaussian quadrature method (Engel, 1980) as ðb 17 a
mB E pffiffiffi G m 1 dm E2 h ð
Selvadurai (1979) showed that infinite integrals involving products of Bessel functions are commonly encountered in the analysis of the axisymmetric problem related to a finite elastic footing resting on a linearly deformable medium. The numerical evaluation of integrals of this type is carried out by representing the integrals as an infinite series bounded by subsequent zeros of the related Bessel functions. Integration, which proceeds by one interval at a time, is carried out using a 15-point Gauss–Legendre quadrature. The 15-point Gaussian formula (Selvadurai, 1979) was used in order to achieve higher accuracy than that obtained from the commonly used four- or eight-point Gaussian formulae. The summation was terminated when the absolute value of the ratio of the area of the last subinterval to the summation of areas of all previous sub-intervals became less than 0.001 (Huang, 1968). Geotechnical Engineering 163 Issue GE2
f ð x Þdx ¼
ba ba ðx þ 1Þ dx f aþ 2 2 1
ð1
where
x ¼ aþ
ba ðx þ 1Þ 2
For the square footing, the values of h/B adopted in this work varied between 0.01 and 25, and the values for the modular ratios E1 /E2 were 0.01, 0.1, 0.2, 0.25, 0.3, 0.5, 0.75, 1, 2.5, 5, 10, 25, 50, 100, 200, 500, 1000 and 2000.
and Equation 13 can be written as
15
wi f ðxi Þ
i¼1
where w i is the weight of the function at the ith point; xi is the position of the function at the ith point; and f (xi ) is the function at the ith point. The values of xi and the weights w i are given by Selvadurai (1979).
18 ¼ J 0 ð0 Þ ¼ 1
15 X
5. COMPUTER PROGRAM To evaluate the elastic surface settlement at the centreline of a square footing on a Burmister two-layer system, the surface deflection factors at the centreline are required. The computer program SDF has been written in the Mathcad language. The first job of the computer program is to calculate the zeros of the Bessel function J1 . The next step is to evaluate the areas between every two successive zeros of the integrand of Equation 15. The zeros of the integrand are determined by the zeros of the pffiffiffi Bessel function J 1 (mB= ð h). After determining the zeros of the integrand (m values corresponding to zeros of Bessel functions J1 ), the integral between each two successive zeros is evaluated using a 15-point Gaussian quadrature as follows
Elastic settlement of square footings on a two-layer deposit
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103
ðb
ba ðx þ 1Þ dx f aþ 2 1 15 b aX ba ðx i þ 1 Þ wi f a þ ¼ 2 i¼1 2
f ð mÞdm ¼
a
19
¼
ba 2
ð1
15 b aX ba ðx i þ 1Þ wi J 1 a þ 2 i¼1 2 B ba p ffiffiffi ðx i þ 1Þ G aþ 3 2 h ð
After calculating the areas by Gaussian quadrature, these areas are summed together, and this summation terminates when the ratio of the absolute value of the last area to the summation of the preceding areas is less than 0.001. To validate the written computer program, it is necessary to compare its results with published results for special cases. Such a special case is given by a square footing on a Boussinesq half-space as given by Equation 1. For this purpose, the case of h/B ¼ 1 with E1 /E2 ¼ 1 was chosen, as it represents a case of a Boussinesq half-space. For a Poisson’s ratio of í ¼ 0.5 (fully saturated clay), Equation 1 becomes
Se ¼
21
0:75qB I5 E
where E ¼ E1 ¼ E2 is the modulus of elasticity of the Boussinesq half-space. According to Das (1985), the influence factor for a point lying on the surface at centreline is I5 ¼ 1.122, so that the surface deflection at centreline of the square footing becomes qB Se ¼ 0:8415 E
22
Using the computer program SDF written in this work, it was
shown that for the case of h/B ¼ 1 with E1 /E2 ¼ 1, Equation 15 yielded a surface deflection factor Fs ¼ 0.845034278. Making use of Equation 11, the surface deflection at the centreline of the square footing becomes w0 ¼ 0.845(qB/E). This is in good agreement with that obtained above from the Boussinesq solution directly. The difference between the two results is due to the approximation of the square footing as an equivalent circle.
6. DESIGN CHARTS To simplify the use of the results of this work in practice, design charts should be developed. These charts are useful in rapid determination of the surface deflection factors to compute the immediate settlement for a flexible square footing based on a two-layer elastic system, taking into account the geometry and strength properties of the system. The behaviour of the surface deflection factor Fs will differ according to the value of the modular ratio and the geometry of the foundation. In practice, two different cases may appear. The first represents the case of a soft top (first) layer over a stiffer bottom (second) layer, reflected through a modular ratio E1 /E2 , 1. The second important case is that of a stiff top layer over a softer bottom layer, reflected through a modular ratio E1 /E2 . 1. For E1 /E2 , 1, Figure 1 shows the surface deflection factor at the centreline of a square footing on a Burmister two-layer elastic system, plotted against the relative layer thickness h/B for values of the modular ratio E1 /E2 of 0.01, 0.1, 0.2, 0.25, 0.3, 0.5, 0.75 and 1. For E1 /E2 .1, Figure 2 shows the surface deflection factor at the centreline of a square footing on a Burmister two-layer elastic system plotted against the relative layer thickness h/B for different values of the modular ratio E1 /E2 of 1, 2.5, 5, 10, 25, 50, 100, 200, 500, 1000 and 2000. It is clear from Figures 1 and 2 that the surface settlement at the centreline of a square footing decreases with increasing
Surface deflection factor, Fs
100
0·01
40 30 20 10
0·1
4 3 2
0·2 0·25 0·3 0·5 0·75
1
E1/E2 ⫽ 1 0·4 0·3 0·2 0·1 0
1
2
3
4
5
6
7
8
9 10
15
20
25
Relative layer thickness, h/B
Figure 1. Surface deflection factor at centreline of a square footing on a Burmister two-layer elastic system for E1 /E2 , 1
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Geotechnical Engineering 163 Issue GE2
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1
E1/E2 ⫽ 1
Surface deflection factor, Fs
2·5 5 0·1
10
0·01
50 100
25
200 500 1000 2000
0·001
0·0001 0
1
2
3
4
5
6
7
8
9 10
15
20
25
Relative layer thickness, h/B
Figure 2. Surface deflection factor at centreline of a square footing on a Burmister two-layer elastic system for E1 /E2 . 1
modular ratio for E1 /E2 . 1, and it increases with decreasing modular ratio for E1 /E2 , 1, especially for the higher values of the relative layer thickness h/B. 7. APPLICATION To show how the developed charts can be used in practice, the case of example 5.6 treated by Bowles (1988) and shown in Figure 3 will be discussed here. Bowles (1988) required the estimation of the immediate settlement at the centreline on the surface of a raft foundation having a rectangular shape of B 3 L ¼ 33.5 m 3 39.5 m subjected to uniform loading q ¼ 134 kPa. The soil is layered clay, with one sand seam from the ground surface to the sandstone bedrock at 11 m. The modulus of elasticity for the top layer extending from 0 to 3 m is E T ¼ 42.5 MPa, and for the second (bottom) layer is EB ¼ 60 MPa. For the sandstone, Bowles (1988) reported Es > 500 MPa. Bowles (1988) estimated the required settlement on the basis of converting the top and bottom layers into an equivalent layer having an average modulus of elasticity of
E ave ¼
The equivalent layer rests on a rigid base. This means that the sandstone was considered rigid. Accordingly, Bowles (1988) reported an immediate settlement at the centreline of 16.5 mm. First, to evaluate the immediate settlement on the centreline of a square footing by making use of the charts developed in this work, the rectangular raft should be converted into an equivalent square raft with the same area and centre. This yields a square raft of 36.4 m side length. Because the ratio of depth of the raft Df ¼ 3 m to its width B ¼ 36.4 m is very small, the effect of foundation depth can be neglected (Das, 2004). To use the chart, the system will be considered as a two-layer system. The first layer is that with Eave ¼ 55 MPa, and the second layer is the sandstone with Es ¼ 500 MPa. To evaluate the immediate settlement by the procedure followed in this work, use can be made of Equation 11. For h/B ¼ 11/36.4 ¼ 0.3022 and E1 /E2 ¼ 55/500 ¼ 0.11, use of the chart in Figure 1 yields Fs ¼ 1.503, and the immediate settlement will be
ð42:5 3 3Þ þ ð60 3 8Þ 11
¼ 55 MPa
w0 ¼
134 3 36:4 3 1000 3 1:503 500 3 1000
¼ 14:6621 mm B ⫻ L ⫽ 33·5 m ⫻ 39·5 m
q ⫽ 134 kPa
The difference between the two results can be attributed to the following. First, Bowles (1988) assumed that the sandstone is completely rigid, whereas in this work the contribution of the sandstone to surface settlement is taken into consideration. Second, the chart in this work is devoted to a Poisson’s ratio of 0.5 for both layers, whereas Bowles (1988) assumed a Poisson’s ratio of 0.35 for the layered clay. Third, the rectangular footing has been converted into an equivalent square one.
Df ⫽ 3 m 3m
Top layer (ET ⫽ 42·5 MPa, ν ⫽ 0·35)
8m Bottom layer (EB ⫽ 60 MPa, ν ⫽ 0·35)
z (m)
ES (sandstone) ⭓ 500 MPa
Figure 3. Elastic settlement below the centre of a flexible foundation for Bowles (1988) example 5.6.
Geotechnical Engineering 163 Issue GE2
To take approximate account of the rectangular shape of the footing, the following correction can be suggested. For the case
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of a Boussinesq half-space, the effect of L/B on the surface deflection factor at the centreline (where L refers to the length and B the width of the rectangular footing) can be obtained from Bowles (1988). For the given raft with L/B ¼ 39.5/33.5 ¼ 1.18 the surface deflection factor I5 (see Equation 1) becomes I5R ¼ 1.207, whereas for the square footing I5S ¼ 1.122. Thus I5R /I5S ¼ 1.0757.
(d) For the case of a stiff top layer over a softer bottom layer, the surface deflection at the centreline of a square footing decreases with increasing modular ratio and increasing relative layer thickness. (e) There is a good agreement between the calculated and measured surface deflection at the centreline of the raft foundation studied by Bowles (1988).
Similarly, according to Equation 1 of Boussinesq, the approximate effect of Poisson’s ratio may be estimated as 2
1 í2a 1 ð0:35Þ : ¼ 2 ¼ 1 17 1 í2u 1 ð0:5Þ where ía is the actual Poisson’s ratio, and íu is the Poisson’s ratio used. Thus the settlement of 14.6621 mm at the centreline obtained on the basis of this work should be multiplied by (1.17 3 1.0757 ¼ 1.2585) to give a more logical immediate settlement of 18.45 mm, which is very close to the measured value of 18 mm as reported by Bowles (1988). It is worth mentioning at this stage that if the immediate settlement is calculated on the basis of the Boussinesq half-space a value of 98 mm will be obtained, which is far from the actual measured settlement of 18 mm reported by Bowles (1988). This indicates the importance of the developed charts. Finally, because of the limitations of the charts for actual ground conditions encountered by practising geotechnical engineers, it is intended in the near future to extend the charts to other (more actual) values of Poisson’s ratio, and to rectangular footings, to encourage the application of the charts in practice. 8. CONCLUSIONS The main conclusions of this work can be summarised as follows. (a) The surface deflection at the centreline of a square footing on a Burmister two-layer soil system obtained from the developed solution of this work is in good agreement, for the case of E1 /E2 ¼ 1, with that obtained from the Boussinesq solution. (b) The surface deflection factor at the centreline of a square footing on a Burmister two-layer elastic system depends on the modular ratio and the geometry of the two-layer soil system. (c) For the case of a soft top layer over a stiffer bottom layer, the surface deflection factor at the centreline of a square footing increases with decreasing modular ratio, especially for higher values of the relative layer thickness h/B.
REFERENCES Bowles JE (1988) Foundation Analysis and Design, 4th edn. McGraw-Hill, New York. Bowles JE (1996) Foundation Analysis and Design, 5th edn. McGraw-Hill, Peoria. Burmister DM (1943) The theory of stresses and displacements in layered systems and application to the design of airport runways. Proceedings, Highway Research Board, 23: 127– 148. Das BM (1985) Advanced Soil Mechanics. McGraw-Hill, New York. Das BM (2004) Principles of Foundation Engineering, 5th edn. Thomson Learning, California State University, Sacramento. Engel H (1980) Numerical Quadrature and Cubature. Acadamic Press, London. Horonjeff R and McKelvey F (1994) Planning and Design of Airports. McGraw-Hill, New York. Huang YH (1968) Chart for determining equivalent single wheel loads. Journal of the Highway Division, ASCE 94(HW2): 115–128. Huang YH (1969) Computation of equivalent single wheel loads using layered theory. Highway Research Record 291: 144– 155. Huang YH (1993) Pavement Analysis and Design. Prentice Hall, Englewood Cliffs, NJ. Kezdi A (1964) Bodenmechanik, Vol. 2. VEB Verlag fuer Bauwesen, Berlin. Poulos HG (1967) Stresses and displacements in an elastic layer underlain by a rough rigid base. Ge´otechnique 17(2): 35–48. Poulos HG and Davis EH (1974) Elastic Solutions for Soil and Rock. Wiley, New York. Razouki SS and Issa W (2001) Displacement due to trapezoidal loading on an elastic layer underlain by a rough rigid base. Proceedings of the 3rd Jordanian Civil Engineering Conference, Amman 1: 312–322. Selvadurai APS (1979) Elastic Analysis of Soil–Foundation Interaction. Developments in Geotechnical Engineering, Vol. 17. Elsevier, Amsterdam. Watson GN (1966) A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge. Yoder E and Witczak M (1975) Principles of Pavement Design, 2nd edn. Wiley, New York.
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