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Ground Improvement (2004) 8, No. 4, 171–177

171

Model testing of strip footings with structural skirts

M. Y. AL-AGHBARI* and Y. E.-A. MOHAMEDZEIN*

*Department of Civil Engineering, Sultan Qaboos University, Sultanate of Oman

Tests on footing models were carried out to study the behaviour of strip footings with structural skirts resting on sand. The use of structural skirts improved the bearing capacity of strip footings by a factor of up to three. The improvement depends on the geometrical and structural properties of the skirts and footing, and the soil characteristics and interface conditions of the soil–skirt–foundation system. To account for these factors, a modi?ed bearing capacity equation was proposed for skirted strip foundations on dense sand by the authors. The predictions of the proposed equation are in excellent agreement with the experimental results. The experimental results showed that the use of structural skirts reduced the settlement by a factor of up to three depending on the skirt depth, geometrical properties of the skirt and footing, soil characteristics and the applied load. The structural skirts modi?ed the load–settlement curve for the footing by increasing the strain at failure to about 22%. ` Nous avons fait des essais sur des modeles d’assises a?n ? ` d’etudier le comportement d’assises a bandes avec des jupes structurales reposant sur du sable. L’utilisation de ? ? ? jupes structurales a ameliore la capacite porteuse des ` ? ? assises a bande par un facteur de 3. L’amelioration depend ? ? ? ? des proprietes geometriques et structurales des jupes et ? des assises ainsi que des caracteristiques du sol et des ` conditions d’interface du systeme sol-jupe-fondation. Pour rendre compte de ces facteurs, les auteurs proposent une ? ? ? equation modi?ee pour la capacite porteuse des fondations ` ? ? a jupe sur du sable dense. Les predictions de l’equation ? ? ? proposee sont en accord parfait avec les resultats experi? ? ? mentaux. Les resultats experimentaux ont montre que ? l’utilisation de jupes structurales reduisait le tassement ` par un facteur allant jusqu’a 3, selon la profondeur de la ? ? ? ? jupe, ses proprietes geometriques et son assise, les car? ? acteristiques du sol et la charge appliquee. Les jupes structurales modi?aient la courbe charge-tassement pour ? ` ` l’assise en augmentant la deformation a la rupture jusqu’a environ 22 %.

Keywords : bearing capacity; settlement; strip footing; structural skirt; sand

Notation

B Bs B9 Df Dr Ds Es F? Nq , N? qo qult SRF s ? ?s ?f ó ? width of foundation without structural skirt width of skirt total width of foundation with skirts ? B + 2Bs depth of foundation base below ground surface relative density of the soil depth of lower edge of skirt below bottom of foundation Young’s modulus of soil skirt factor bearing capacity factors ultimate bearing capacity for footing without skirts ultimate bearing capacity for footing with skirts settlement reduction factor settlement unit weight of soil angle of friction of sides of skirt angle of friction of base of foundation pressure applied by footing angle of internal friction of soil

Introduction

Reinforcements to improve the performance of soils have been used in different geotechnical engineering applications (Bell, 1993). In shallow foundations both horizontal and

(GI 3186) Paper received 8 August 2003; last revised 7 January 2004; accepted 20 April 2004 Delivered by ICEVirtualLibrary.com to:

vertical reinforcements have been used to improve the bearing capacity and reduce the anticipated settlement (Mahmoud and Abdrabbo, 1989; Das, 1999). Flexible horizontal reinforcements such as geotextiles are useful for improving bearing capacity, but they require a large amount of settlement to mobilise the shear resistance. Mahmoud and Abdrabbo (1989) used vertical non-extensible reinforcement rods placed within a distance of 2B from the edge of the footing. The vertical reinforcement improved the bearing capacity against general shear failure. Improvements in other types of shear failure, such as local and punching shear failure, are not reported. Another variation of the vertical reinforcement, called structural skirts, is found to improve resistance to all types of bearing failure and to reduce settlement (Al-Aghbari, 2002; Al-Aghbari and Mohamedzein, 2004). Structural skirts are rigid steel plates ?xed to the edge of the foundation and penetrating a suf?cient depth into the soil. Structural skirts have been used for a considerable time in offshore structures and other situations where water scour may be a problem (Bransby and Randolph, 1998; Watson and Randolph, 1998; Hu et al., 1999). Structural skirts are preferred for the support of huge gravity offshore structures, even in soft marine deposits, because of their short installation time, economic feasibility and satisfactory performance under cyclic loading. Hu et al. (1999) studied a circular skirted offshore foundation on non-homogeneous soil. Watson and Randolph (1998) reviewed the performance of skirted foundations in calcareous soil.

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Al-Aghbari and Mohamedzein The use of such structural skirts in conjunction with conventional shallow structural foundations has not been widely employed, nor have the improvements in bearing capacity occurring from their use been investigated in detail. Al-Aghbari and Mohamedzein (2004), using a laboratory model (50 mm wide and 150 mm long), studied the bearing capacity of a strip footing with structural skirts. They modi?ed the classical Terzaghi bearing capacity equation (Terzaghi, 1943) to account for the effect of the skirt. The proposed equation is based on a similar failure surface to that given by Terzaghi (Fig. 1). The proposed equation is 1 qult ? ?? Df ? Ds ?Nq ? B9?N? F? 2 (1) footing on sand. The footing model was 100 mm square. The use of structural skirts was found to reduce the settlement to about 10% of the settlement of the footing without skirts. A settlement reduction factor (SRF) was proposed, and takes the form SRF ? e??0 012ó Ds = B9?

:

(3)

where F? is the skirt factor, Df is the depth to the foundation base below ground level, Ds is the depth to the lower edge of the skirt below the foundation base, B9 is the total foundation width with skirts (? B + 2Bs ), and Bs is the skirt thickness. The skirt factor (F? ) is introduced into the second part of the general equation to account for all the characteristics of the structural skirts, the soil, the foundation and the loading that in?uence the ultimate bearing capacity of the foundation. No factor is included in the ?rst part of the general equation because the effect of the skirt can be accounted for by the skirt depth. The skirt factor (F? ) depends on the angle of internal friction of the soil (?), the angle of friction of the base of the foundation (?f ), the angle of friction of the sides of the skirt (?s ), the skirt stiffness, and the compressibility of the soil between the skirts. Based on the results of their laboratory model Al-Aghbari and Mohamedzein (2004) proposed the following expression for the skirt factor: tan ?9 F? ? 1:15 0:4 ? 0:6 tan ?f " # Ds tan ?s ? 0:37 3 ?1:2 ? 0:002Dr ? 3 0:57 ? 0:1 B9 tan ?f (2) where Dr is the relative density of the soil between the skirts. In developing the above equation the parameters of interest were varied as shown in Table 1. Al-Aghbari (2002) conducted model tests to study the effect of structural skirts on the settlement of a square

B′ Df Ds

where ó is the pressure applied on the footing (kN/m2 ), Ds is the skirt depth, and B9 is the width of footing with structural skirts. Equations (1) to (3) are based on model tests. The scale and shape effects on the results of these models are investigated in this study by using a model strip footing with a width that is 1.2 and 2.4 times the widths of the square and strip models respectively. The paper also presents an insight into the settlement behaviour of footings on sand with and without structural skirts.

Experimental

Test set-up and procedures

The foundation was tested in a tank formed by a very heavily reinforced rigid steel frame and toughened glass sides. This test set-up was intended to simulate the plane strain condition beneath a strip footing. The main concern in simulating the plane strain condition was to avoid the side effects of the test tank. This was accomplished by maintaining the rigidity of the test tank (Ko and Davidson, 1973), and by placing the model footing at a reasonable distance from the sides of the tank. The inner dimensions of the tank were 2 m length, 1.4 m depth and 0.3 m width (Fig. 2). The foundation was placed centrally across the tank and was manufactured from rigid smooth steel plate. It was 0.12 m wide and 0.3 m long. The tank and the footing dimensions are similar to those used by others (e.g. Ko and Davidson, 1973; Kirkpatrick et al., 1987; Yetimoglu et al., 1994; Zadroga, 1994). As the length and the depth of the tank are more than 10 times the width of the footing, the boundary effect on the results was considered small. In fact, during the tests it was observed that the failure surface on the sides of the footing did not reach the edge of the tank, indicating that the boundary effect on the tests was likely to be insigni?cant. Similar conclusions were reported by Yetimoglu et al. (1994). The central vertical load was applied by means of a motorised 10 t capacity screw jack operated to apply a constant rate of displacement of 12 mm/h. The applied loads

q

γ(Df

Ds)

2000

Fig. 1. Bearing capacity failure mechanism in soil under continuous foundation with structural skirt subjected to vertical central load. Based on Terzaghi (1943) assumptions

1400

300 120

Table 1. Range of different parameters used in developing equation (2) (after Al-Aghbari and Mohamedzein, 2004) Parameter Angle of friction of base of foundation, ?f (degrees) Angle of friction of sides of skirt, ?s (degrees) Relative density of soil between skirts, Dr (%) Skirt depth/footing width, Ds /B9 Skirt thickness, Bs (mm) Range 27–37 15–37 12–89 0.4–1.6 0.25–6 Fig. 2. Dimensions of the tank Delivered by ICEVirtualLibrary.com to:

All dimensions in mm

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Model testing of strip footings with structural skirts were measured using two load cells each of 40 kN capacity, and the displacements of the foundation were measured by four LVDTs. Tests were carried out on the foundation without skirts and with skirts. The skirts were rigidly connected to the sides of the foundation; skirt thicknesses of 6 mm and 12 mm were used. The depth of the skirt was also varied up to 0.18 m. At the beginning of each test the glass side plates of the tanks were cleaned with acetone. To ensure lubrication of the sides of the tank, silicon grease and rubber membranes were used. The membranes were greased and left for at least 1 h to allow uniform spreading of the grease over the surface of the membranes. The membranes were then placed on the glass symmetrically along the centreline of the foundation. Care was taken to remove trapped air bubbles between the glass wall and the membrane. This step was intended to further reduce the boundary effects on the test results. After the tank was prepared, the tank was then ?lled with sand using the sand spreader method (Butter?eld and Andrawes, 1970). This is a well-known technique for controlling the porosity (i.e. relative density) of the sand during deposition. By controlling the height of fall and the rate of deposition any porosity within the maximum/minimum porosity range can be obtained (Walker and Whitaker, 1967; Butter?eld and Andrawes, 1970; Kirkpatrick et al., 1987). Furthermore, the method produces beds of sand that are homogeneous and reproducible over a wide range of porosities. In this method a sand spreader was placed on the top of the tank and ?lled with sand. An arrangement was made to keep the sand falling height constant, leading to more homogeneous sand (Al-Aghbari, 1999). After the sand had reached the required height at the base of the foundation or the skirt, it was levelled with a levelling apparatus. The foundation without skirts was placed very carefully in the tank on top of pre-placed sand, and more sand was deposited as mentioned earlier. When the sand reached the required level, the sand spreader was stopped and the sand surface was levelled. For the foundations with skirts, the skirts were placed ?rst in a de?ned position vertically on the top of the pre-placed sand and were connected together by two rods to keep them at a distance equal to the width of the foundation. After the skirts were placed in the tank, sand was deposited between the skirts using the same procedure as outlined above. Then the sand spreader was stopped when the required sand level reached the level of foundation base, and the surface of the sand between the skirts was levelled. The screw rods connecting the skirts were removed, and the foundation was placed between the skirts. The skirts were ?xed to the foundation by eight bolts on each side. Care was taken in placing and bolting the skirts in such a way that no disturbance occurred. An open box was placed on the top of the foundation. Then the sand spreader was switched on to spread the sand to the required sand level. After that the sand spreader was stopped, the box was removed, and the sand surface was levelled. The loading rig assembly was lifted from its seating using a crane and taken over the tank centre. The loading blades were connected to the loading bars of the rig and were placed on the centre of the foundation. The load cells and LVDTs were placed and connected to the data logger. The data logger was connected to a computer for recording all readings. The readings of all load cells and LVDTs were scanned before any loading. After the initial scanning was ?nished, the motor of the loading rig was switched on at a displacement rate of 12 mm/h. The data logger was set for automatic scans every 1 min. At the end of the test, the load on the foundation was released by operating the jack manually. Fig. 3 shows a complete set-up of the apparatus. It should be noted that the method of placing the skirt in the sand is intended to reduce the disturbance of the sand, and to have the same density as that for the footing without skirts. However, in practice skirts can be driven. Driven skirts are likely to densify loose sand deposits, improve the bearing capacity and reduce the settlement by more than what is reported in this study. Thus the results reported here can be considered conservative for driven skirts in loose sand.

Properties of the soil

The sand ?lling the tank was Leighton Buzzard sand, which consists of uniform rounded particles. The dominant mineral composition is quartz. The particle size distribution is shown in Fig. 4. The sand has a coef?cient of uniformity (Cu ) of 1.45 and a coef?cient of curvature (Cc ) of 0.92. The

7 Chain 6 Chain 8

4

3

5

2

Sand

1

Fig. 3. General layout of the foundation apparatus: 1, rigid steel frame; 2, glass plate (12.5 mm); 3, supports to the glass sides; 4, sand spreader motor; 5, foundation; 6, 10 t screw jack; 7, foundation motor and gearbox; 8, loading system

100 90 80 Percentage finer by weight 70 60 50 40 30 20 10 0 0·01 0·1 1 10 Particle diameter: mm

Fig. 4. Grain size distribution of Leighton Buzzard sand Delivered by ICEVirtualLibrary.com to: IP: 131.251.133.28 On: Thu, 16 Jun 2011 10:06:39

173

Al-Aghbari and Mohamedzein minimum and maximum porosities of the sand are 34.8% and 46.4% respectively, and were obtained in accordance with BS 1377 (British Standards Institution, 1990). The speci?c gravity was found to be 2.67. The sand was placed in the tank by an air-activated spreader as described by Butter?eld and Andrawes (1970). The relative density of the placed sand was 86%, corresponding to a unit weight of 16.54 kN/m3 . The peak angle of friction for the dense sand was determined over the con?ning stress in the range 40– 200 kN/m2 using the shear box, triaxial and plane strain tests, and was found to be 458, 468 and 498 respectively. Fig. 5 shows the results of triaxial tests on this sand. The stress– strain and volume change curves are typical for dense sand. It should be noted that skirts can be used to improve the bearing capacity and reduce the compressibility of loose or soft soils. The use of the dense sand was intended to study the factors that in?uence the behaviour of skirted footings, such as the dimensions of the footing and skirts, friction between the soil and the skirts, and friction between the footing and the soil. The angle of friction between the dense sand and the smooth steel skirt was determined in the shear box over the same stress range and found to be 278. The skirts were also treated to make them smoother or rougher. The angles of friction between the dense sand and the treated steel skirt were found to vary from 358 to 378 depending upon the surface treatment employed for the skirts.

Results and discussion

Selection of value of angle of internal friction

For the comparisons between the experimental results and bearing capacity equations (e.g. equation (1)), a representative value of the peak angle of friction for the soil (?9) is required to be identi?ed. Published results (Ko and Davidson, 1973; Kirkpatrick et al., 1987) indicated that the classical bearing capacity equations underestimate the bearing capacity when the values of ?9 obtained from triaxial tests are used, and overestimate the bearing capacity when the values of ?9 obtained from plane strain tests are used. Meyerhof (1963) and Hansen (1970) used plane strain values, and suggested that these may be up to 10% higher than the corresponding triaxial test values. Based on the above facts an operational value of 47.58 was used, which is the average of the triaxial and plane strain test values.

Tests on foundations with skirts

Bearing capacity

The behaviour of strip footings with structural skirts resting on sand was studied for different combinations of the factors such as footing depth, skirt depth, skirt thickness and skirt side friction, as shown in Table 2. The ratio of the measured bearing capacity of the footings with skirts to that without skirts is also presented in Table 2. The table shows an improvement in bearing of 1.72 to 3.12, and the improvement depends on the factors given above. For instance, the improvement decreases with increased footing depth; the highest improvement is found for surface footings. By contrast, the improvement increases with an increase in the skirt depth. Making the skirt rigid by increasing the thickness of the skirt will also enhance the bearing capacity of the footing. The improvement in bearing capacity is also more signi?cant for smooth footings than for rough footings. The expected improvement in the bearing capacity can be quanti?ed in terms of a lumped parameter called the skirt factor, as shown in equations (1) and (2). A comparison is made between the predictions of equations (1) and (2) and the measured ultimate bearing capacity, as shown in Table 3 and Fig. 6. It is clear that the predicted ultimate bearing capacity (qult,predicted) and measured ultimate bearing capacity (qult,measured ) are approximately equal with an error of less than 5%. This implies that the proposed equations for a footing 50 mm wide and 150 mm long are valid for a relatively larger model footing (120 mm wide and 300 mm long).

1400 Confining pressure 1200 Deviator stress: kN/m2 1000 800 600 400 200 0 0 5 10 15 Axial strain: % (a) 20 25 40 kN/m2 100 kN/m2 150 kN/m2 200 kN/m2

10

Volumetric strain: %

Confining pressure 40 kN/m2 100 kN/m2 8 150 kN/m2 200 kN/m2 6 4 2 0 2 0 5 10 15 20 25 Axial strain: % (b)

Settlement

Foundation design requires two aspects of behaviour to be taken into account: the ultimate bearing capacity of the soil under the foundation, and the acceptance limits for foundation settlement. Excessive settlement may cause distress to the superstructure. Bearing capacity failure is usually initiated after the occurrence of a large amount of settlement. Based on the results reported by Vesic (1963, 1973), the ultimate bearing capacity of a circular footing may occur at footing settlements of 4–10% of the footing width for the case of general bearing capacity failure. For rectangular footings, general bearing capacity failure may occur at settlements of 10–15%. For both circular and rectangular footings the ultimate bearing capacity is mobilised at settlements of 15–25% for the cases of local and punching shear failures. In this study general shear failure is observed for strip footings with and without structural skirts. The results

Fig. 5. Leighton Buzzard sand: (a) stress–strain relation; (b) volume change characteristics Delivered by ICEVirtualLibrary.com to:

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Model testing of strip footings with structural skirts

Table 2. Comparison of bearing capacity for foundation with and without structural skirts Test no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Df /B9 0 0 0 0 0 0 0 0.454 0.454 0.454 0.454 0.454 0.416 0.416 0.416 Skirt thickness, Bs : mm 6 6 6 6 6 6 6 6 6 6 6 6 12 12 12 Ds /B9 0.04 0.454 0.454 0.91 0.91 0.454 0.91 0.454 0.91 0.454 0.91 1.36 0.416 0.833 1.25 Skirt side friction angle, ?s (degrees) 35 27 35 35 27 37 37 27 27 37 37 37 37 37 37 Ratio qult, measured with skirts qult, footing without skirts 1.87 2.18 2.45 3.09 2.76 2.49 3.12 1.72 2.01 1.86 2.21 2.60 1.93 2.39 2.76

Table 3. Comparison of measured and predicted bearing capacity

Applied stress: kN/m2

2000

Test no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

qult,predicted : kN/m2 971 1120 1241 1538 1417 1275 1572 1376 1673 1531 1828 2122 1620 1916 2213

qult,measured : kN/m2 945 1103 1239 1564 1397 1260 1575 1312 1576 1417 1683 1979 1468 1816 2098

Ratio qult,predicted qult,measured 1.027 1.016 1.002 0.983 1.015 1.012 0.998 1.048 1.062 1.080 1.086 1.072 1.103 1.055 1.054

1500

Df /B = 0 0·5 1 1·5 2

1000

500

0 0 5 10 15 s/B: % 20 25 30

Fig. 7. Load–settlement relationship for footings without structural skirts

2000

Applied stress: kN/m2

1500 Test no. 1000 1 2 3 4 5 6 7

2500

Predicted ultimate bearing capacity: kN/m2

2000

500

0 1500 0 5 10 15 s/B: % 20 25 30

1000

Fig. 8. Load–settlement relationship for surface footings with structural skirts

500 500

1000

1500

2000

2500

Experimental ultimate bearing capacity: kN/m2

Fig. 6. Comparison between measured and predicted bearing capacity

are presented in Fig. 7 for footings without structural skirts and in Figs 8 and 9 for footings with structural skirts. The ?gures show that the ultimate bearing capacity for strip footings without structural skirts generally occurs at settle-

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ment of 13–19%, and at settlements of 18–22% for footings with structural skirts. For both cases the values are higher than those reported by Vesic for other shapes of footings. The structural skirts reduced settlement by a factor of up to 3. The settlement reduction factor (SRF) depends on the dimensions of the footing and the skirt, the friction between the skirt and the soil, and the applied stress on the footing. To establish a correlation between the above factors and SRF, values of SRF were obtained at ?ve different stress levels for each test on surface footings and at seven different stress levels for each test on the shallow footings. Thus a total of 75 sets of data were used in the analysis. Following 175

Al-Aghbari and Mohamedzein

2500

Applied stress: kN/m2

2000

Test no. 8 9 10 11 12 13 14 15

1500

Young (1987) and Fakher and Jones (1996), dimensional analysis was performed to show the scale effect on the bearing capacity and settlement of a strip footing with structural skirts. Important factors that affect the bearing capacity are qult ? f ? qo , B9, ?, Df , Ds , ó ? (5)

1000

500

0

0

5

10

15 s/B: %

20

25

30

where qo is the bearing capacity for a footing without structural skirts. The following dimensionless parameters (called ? terms) can be considered: qult qo Df Ds ó (6) ?? , , , B9? B9? B9 B9 B9? Application of the results of the model test to the prototype require that each of the ? terms be the same for the model and the prototype. For example, the following requirements must be satis?ed: qult qult Df Df ? ; ? ... (7) B9? p B9? m B9 p B9 m It is clear from these equations that, in addition to the geometric scale factor related to footing and skirt dimensions, other factors related to soil type, soil placement and compaction and stress level must be considered. If the factors related to the soil and stress level are the same for the model and the prototype, then the results from smallscale tests can be applied to the prototype after taking the geometrical effect into account. Similarly, dimensional analysis for settlement can give the following dimensionless terms: s ó ?B9 Df Ds , (8) , , ?? B9 Es Es B9 B9 where s is settlement, and Es is the Young’s modulus of the soil. As for to the bearing capacity, this equation indicates that the scale effect for settlement depends on geometrical, soil and stress level factors. For the laboratory models used in this study the soil and stress factors are similar. Therefore the results obtained from these tests are comparable. However, for actual footings the soil placement and compaction, details of skirt construction and stress level will be different from the laboratory models. Further investigations using full-scale tests or numerical models are required to generalise the results of the model tests to actual footings.

Fig. 9. Load–settlement relationship for shallow footings with structural skirts

similar principles to those outlined by Montgomery and Runger (1999) for multiple regression analysis, the following equation is proposed to predict the settlement reduction factor: p?????????? : SRF ? ?1:3 ? 0:011?s ?e?0 035ó Ds = B9 (4) where ó is the pressure applied on the footing (kN/m2 ), Ds is the depth of the skirt, B9 is the width of footing with structural skirts, and ?s is the skirt side friction angle in degrees. The values of SRF obtained from equation (4) are compared with the measured values in Fig. 10. The agreement between the proposed equation and the measured values is quite good (with R2 ? 0.87).

Scale effect

The results presented in the previous section were based on the results of model footings. Because of the scale effect these results will not be applicable to actual footings (Aiban and Znidarcic, 1995). The scale effect can be explained by dimensional analysis. Following approaches proposed by

1·00

0·80

Conclusions

Experimental SRF 0·60

A model test was conducted to study the behaviour of skirted strip foundations bearing on sand. The results from this investigation can be summarised as follows: (a) The structural skirts improved the bearing capacity by a factor of up to 3.12. (b) The structural skirts reduced settlement by a factor of up to 3. (c) The ultimate bearing capacity for strip footings without structural skirts can occur at a settlement of 17–19% of the width of the footing. For footings with structural skirts the corresponding settlement is up to 22%. (d) Based on the results of tests on model footings, equations were proposed for ultimate bearing capacity and settlement reduction factor for strip footings with structural skirts. The in?uence of footing size and

0·40

D f /B = 0

D f /B = 0·5 0·20

0 0 0·20 0·40 0·60 0·80 1·00 Calculated SRF

Fig. 10. Relationship between proposed empirical SRF and experimental SRF Delivered by ICEVirtualLibrary.com to:

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Model testing of strip footings with structural skirts construction details of skirts on these equations has not been investigated. Thus further studies are needed to extend these equations to actual footings.

References

Aiban S. A. and Znidarcic D. (1995) Centrifugal modelling of bearing capacity of shallow foundations on sand. Journal of Geotechnical Engineering, ASCE, 121, No. 10, 704–712. Al-Aghbari M. Y. (1999) Bearing Capacity of Shallow Strip Foundation with Structural Skirts Resting on Dense Sand. PhD thesis, University of Strathclyde, Glasgow, UK. Al-Aghbari M. Y. (2002) Settlement of shallow square foundation with structural skirts resting on sand. Proceedings of the 2nd International Conference on Geotechnical and Geoenvironmental Engineering in Arid Lands, Riyadh, 189–194. Al-Aghbari M. Y. and Mohamedzein Y. E.-A. (2004) Bearing capacity of strip foundations with structural skirts. Journal of Geotechnical and Geological Engineering, 22, No. 1, 43–57. Bell F. G. (1993) Engineering Treatment of Soils. E&FN Spon, London. Bransby M. F. and Randolph M. F. (1998) Combined loading of ? skirted foundation. Geotechnique, 48, No. 5, 637–655. British Standards Institution (1990) British Standard Methods of Test for Soils for Civil Engineering Purposes. BSI, Milton Keynes, BS 1377. Butter?eld R. and Andrawes K. Z. (1970) An air activated sand ? spreader for forming uniform beds. Geotechnique, 20, No. 1, 97–100. Das B. M. (1999) Principles of Foundation Engineering. PWS Publishing, Paci?c Grove, CA. Fakher A. and Jones C. J. F. P. (1996). Discussion: Bearing capacity of rectangular footings on geogrid-reinforced sand. Journal of Geotechnical Engineering, ASCE, 122, No. 4, 326–327. Hansen J. B. (1970) A Revised and Extended Formula for Bearing Capacity. Bulletin No. 28, Danish Geotechnical Institute, Copenhagen, pp. 5–11. Hu Y., Randolph M. F. and Watson P. G. (1999) Bearing response of skirted foundation on nonhomogeneous soil. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 125, No. 11, 924–935.

Kirkpatrick W. M., Andrawes K. Z. and Wong F. K. (1987) Contact pressures and failure mechanisms of square footings in sand. Proceedings of the 9th Asian Geotechnical Conference, Bangkok, 6, pp.89–104. Ko H.-Y. and Davidson L. W. (1973) Bearing capacity of footings in plane strain. Journal of the Soil Mechanics and Foundation Engineering Division, ASCE, 99, No. SM1, 1–22. Mahmoud M. A. and Abdrabbo F. M. (1989) Bearing capacity tests on strip footing resting on reinforced sand subgrades. Canadian Geotechnical Journal, 26, No. 1, 154–159. Meyerhof G. G. (1963) Some recent research on the bearing capacity of foundations. Canadian Geotechnical Journal, 1, No. 1, 16–26. Montgomery D. C. and Runger G. C. (1999) Applied Statistics and Probability for Engineers. John Wiley & Sons, New York. Terzaghi K. (1943) Theoretical Soil Mechanics. John Wiley & Sons, New York. Vesic A. S. (1963) Bearing capacity of deep foundations in sand. Highway Research Record No. 39, National Academy of Sciences, Washington, DC, pp. 112–153. Vesic A. S. (1973) Analysis of ultimate loads of shallow foundations. Journal of the Soil Mechanics and Foundation Engineering Division, ASCE, 99, No. SM1, 45–73. Walker B. P. and Whitaker T. (1967). An apparatus for forming ? uniform beds of sand for model foundation tests. Geotechnique, 17, No. 2, 161–167. Watson P. G. and Randolph M. F. (1998) Skirted foundations in calcareous soil. Proceedings of the Institution of Civil Engineers, Geotechnical Engineering, 131, No. 3, 171–179. Yetimoglu T., Wu J. T. H. and Saglamer A. (1994) Bearing capacity of rectangular footings on geogrid-reinforced sand. Journal of Geotechnical Engineering, ASCE, 120, No. 12, 2083–2099. Young D. F. (1987) Similitude, modeling and dimensional analysis. In Handbook on Experimental Mechanics (ed. A. S. Kobayashi), Prentice Hall, Englewood Cliffs, NJ, pp. 621–658. Zadroga B. (1994). Bearing capacity of shallow foundations on noncohesive soils. Journal of Geotechnical Engineering Division, ASCE, 120, No. 11, 1991–2008.

Discussion contributions on this paper should reach the editor by 1 April 2005

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- Undrained Limit Analyses for Combined Loading of Strip Footings on Clay