THE UNIVERSITY OF ADELAIDE SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING

GEOTECHNICAL ENGINEERING DESIGN III

M. B. Jaksa

DESIGN OF SHALLOW FOUNDATIONS References: Bowles, J. E. (1988). Foundation Analysis and Design, 4th ed., McGraw Hill, 1004p.

Coduto, D. P. (1994). Foundation Design - Principles and Practices, Prentice Hall, 796p. Das, B. M. (1995). Principles of Foundation Engineering, 3rd ed., PWS Publ. Co., 828p. Fang, H.-Y. (ed.) (1991). Foundation Engineering Handbook, 2nd ed., Chapman and Hall, 923p. Fraser, R. A. and Wardle, L. J. (1976). Numerical Analysis of Rectangular Rafts on Layered Foundations. Géotechnique, Vol. 26, No. 4, pp. 613–630. Tomlinson, M. J. (1986). Foundation Design and Construction, 5th ed., Longman, 842p. Zhang, B. Q. and Small, J. C. (1991). Finite Layer Analysis of Soil-Raft Interaction. Research Report, University of Sydney, 66p.

1. INTRODUCTION As mentioned previously, the design of foundations must include assessment of at least the following three criteria: (i) bearing capacity; (ii) settlement; and (iii) construction and economic feasibility. 2. DESIGN OF PAD FOOTINGS The design of pad, or spread, footings involves the following procedure: 1. Estimate the preliminary allowable bearing pressure, qall

* , from Table 1.1 (Bearing Capacity of

Shallow Foundations Lecture Notes) based on the underlying foundation material, such that:

qq

FSallu* = (2.1)

2. Determine the width of a square footing, B, or the diameter of a circular footing, D, dependent on qall

* , such that:

BP

qall

= * (square footings), DP

qall

= 4

π * (circular footings) (2.2)

where: P is the vertical column load;

3. Perform a bearing capacity analysis on the smallest of the proposed footings and calculate the maximum allowable bearing pressure, qall(bearing);

4. Perform a settlement analysis on the largest of the proposed footings and calculate the maximum allowable bearing pressure, qall(settlement), that keeps the settlement within tolerable limits;

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5. Determine the design allowable bearing pressure, qall , which is the lower of qall(bearing) and qall(settlement).

Example 2.1: A proposed building is to have column loads which vary between 130 and 1000 kN. The underlying foundation material is a medium-dense sand. If the allowable total settlement is 15 mm, determine the allowable bearing pressure for 0.5 m deep, square footings. (Assume that c = 0, φ = 35° and γ = 18 kN/m3, νs = 0.5, Es = 50 MPa). Solution: 1. From Table 1.1 (Bearing Capacity of Shallow Foundations Lecture Notes): for medium-dense

sand, qu = 300 kPa. Choosing FS = 3:

qall* = =300

3100 kPa

2. BP

qall

= = ≈*

130

1001.1 m ;

BP

qall

= = ≈*

,.

1 000

10023 m

3. Determine allowable bearing pressure for 1.1 m square footing. Using Terzaghi’s equation: q cN qN BNu c q= + +1.3 0.4γ γ = 0 + 18 × 0.5 × 41.4 + 0.4 × 18 × 1.1 × 42.4 = 0 + 372.6 + 335.8

∴ qu = 708.4 kPa.

qall bearing( ) = =708.4

3236.1 kPa

4. Determine settlement for 3.2 m square footing. Using Equation (4.1) from Loading Induced

Stresses and Displacements Lecture Notes:

( )sqB

EIi s= −1 2ν . From Loading Induced Stresses and Displacements Lecture Notes: Is = 0.82

for average settlement of a rigid square footing and Is = 0.95 for average settlement of a flexible square footing. Taking the average of these 2 values, Is = 0.90.

Rearranging to solve for q:

( ) ( )qs E

B Ii

s

=−

= ×−

= =1

15 50

3 12 2ν ν,200 0.90.347 MPa 347 kPa

5. Therefore, qall is smallest of qall(bearing) and qall(settlement), and rounding down to the nearest 25 mm: qall = 225 kPa.

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3. DESIGN OF STRIP FOOTINGS The structural design of strip, or continuous, footings is very similar to that of pad footings. The width of the strip footing is chosen to satisfy bearing capacity and settlement criteria, and the depth of the footing is determined to satisfy beam shear requirements (Warner et al., 1989). The flexural steel reinforcement is evaluated assuming that the strip footing undergoes one-way bending, a one metre length being designed as a double cantilever. 4. DESIGN OF MAT (RAFT) FOOTINGS Mat foundations are designed so that bearing capacity failure does not occur and so that settlements, particularly differential settlements, are maintained within acceptable limits. Tables 4.1 and 4.2 show typical acceptable differential settlements and deflection ratios (∆ / L), respectively. Table 4.1 Acceptable differential settlements in buildings (mm). (Source: Bowles, 1988.)

Criterion Isolated Footings Mat Foundations Angular distortion (cracking) 1 / 300 1 / 300

Greatest differential settlement: Clays 40 40 Sands 30 30

Maximum settlement: Clays 60 60 − 100 Sands 50 50 − 70

Table 4.2 Acceptable deflection ratios (∆∆∆∆ / L). (Source: Das, 1995.)

Category of Potential Damage ∆∆∆∆ / L Danger to machinery sensitive to settlement 1 / 750 Danger to frames with diagonals 1 / 600 Safe limit for no cracking of buildings 1 / 500 First cracking of panel walls 1 / 300 Difficulties with overhead cranes 1 / 300 Tilting of high rigid buildings becomes visible 1 / 250 Considerable cracking of panel and brick walls 1 / 150 Danger of structural damage to general buildings 1 / 150 Mat, or raft, foundations rarely fail as the result of bearing capacity failure because of their relatively large width. However, bearing capacity might be important with cohesive soils, especially if undrained conditions apply. Bearing capacity is assessed using the techniques detailed in the Bearing Capacity of Shallow Foundations lecture notes. The design of a mat foundation built on a soil involves both the behaviour of the supporting soil and of the mat itself, and so a full soil-structure interaction analysis is necessary. Often, sophisticated computer packages are used to model the behaviour of the concrete mat on the underlying soil. The

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simplest way to account for the deformation of the soil is to assume that the pressure at any point on the surface of the soil is proportional to the deformation of the soil at that point. This results in the Winkler model (1867) which, in effect, treats the soil as a series of springs, as shown in Figure 4.1.

Figure 4.1 Winkler spring model. While affording a relatively simple means of obtaining the deformation response of the soil, the Winkler model neglects the interaction of one spring with another and, therefore, does not treat the underlying soil as a true continuum. An alternative and improved approach is to treat the soil as an elastic continuum. Fraser and Wardle (1976) used finite elements to model rectangular rafts of dimensions l × b (such that l > b), supported by a homogeneous isotropic layer of thickness d resting on a rigid base, as shown in Figure 4.2.

Figure 4.2 Rectangular raft foundation on a homogeneous layer. (Source: Fraser and Wardle, 1976.)

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The authors generated a series of graphical solutions, as shown in Figures 4.3 to 4.11. The charts are based on the following relationships:

• Stiffness factor, K: ( )( )K

E v t

E v br s

s r

=−−

4

3

1

1

2 3

2 3 (4.1)

where: Er is the modulus of elasticity of the raft;

νr is the Poisson’s ratio of the raft; Es is the modulus of elasticity of the soil; νs is the Poisson’s ratio of the soil; and t is the thickness of the raft.

For flexible rafts, K → 0, and for rigid rafts, K → ∞.

• Settlement, ρ: ( )

ρ =−

pbv

EI

s

s

1 2

(4.2)

where: p is the applied uniform pressure on the raft;

I is the influence factor for settlement.

Note: ρ and I have the following typical subscripts: A, B and C – associated with central settlement (point A, Fig. 4.3), mid-edge settlement (point B, Fig. 4.3) and C corner settlement (point C, Fig. 4.3), respectively; and AB and AC associated with differential settlement between the centre and mid-edge, and the centre and corner, respectively.

Figure 4.3 Settlement influence factors, I. (Source: Fraser and Wardle, 1976.)

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• Maximum bending moment, m: m plb M= (4.3) (per unit width)

where: M is the bending moment influence factor of the raft.

Note: MAB refers to bending about the axis AB. Correction factors, S (settlement) and R (bending moment), are applied to raft foundations for the effect of finite layer depth, d, such that: ρ ρ= S si (4.4)

where: ρsi is the settlement based on a semi-infinite soil mass. (S = 1 for a semi-infinite soil mass).

m Rmsi= (4.5)

where: msi is the bending moment based on a semi-infinite soil mass. (S = 1 for a semi-infinite soil mass).

Figure 4.4 Bending moment influence factors, M. (Source: Fraser and Wardle, 1976.) Example 4.1 Consider a square raft foundations of dimensions 10 m × 10 m and 0.5 m thick, subject to a uniform load of 100 kPa resting on a soil layer of thickness d = 40 m. The raft properties are as follows: Er = 15,000 MPa, νr = 0.2 and the soil properties are: Es = 81.9 MPa, νs = 0.3. Determine (a) the central settlement, (b) the differential settlement between the centre and the mid-edge of the raft, and (c) the maximum bending moment.

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Figure 4.5 Settlement correction factor, S (l/b = 1): (a) ννννs = 0; (b) ννννs = 0.3; (c) ννννs = 0.5. (Source: Fraser and Wardle, 1976.)

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Figure 4.6 Settlement correction factor, S (l/b = 2): (a) ννννs = 0; (b) ννννs = 0.3; (c) ννννs = 0.5. (Source: Fraser and Wardle, 1976.)

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Figure 4.7 Bending moment correction factor, R (l/b = 1): (a) ννννs = 0; (b) ννννs = 0.3; (c) ννννs = 0.5. (Source: Fraser and Wardle, 1976.)

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Figure 4.8 Bending moment correction factor, R (l/b = 2): (a) ννννs = 0; (b) ννννs = 0.3; (c) ννννs = 0.5.

(Source: Fraser and Wardle, 1976.)

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Figure 4.9 Settlement influence factors, IA (l/b ≤≤≤≤ 5): (a) d/b = ∞∞∞∞; (b) d/b = 1. (Source: Fraser and Wardle, 1976.)

Figure 4.10 Differential settlement influence factors, IAB , IAC , IAD , (3 ≤≤≤≤ l/b ≤≤≤≤ 5): (a) d/b = ∞∞∞∞; (b) d/b = 1. (Source: Fraser and Wardle, 1976.)

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Figure 4.11 Bending moment influence factors, M, (l/b ≤≤≤≤ 5): (a) d/b = ∞∞∞∞; (b) d/b = 1.

Solution: Firstly, calculate K:

From Equation (4.1): ( )( )

( )( )K

E v t

E v br s

s r

=−−

= ×−

−=4

3

1

1

4

3

15 000 1 500

1 10 000

2 3

2 3

2 3

2 3

,

,

0.3

81.9 0.20.0289

K is close to 0 ∴flexible raft. (a) From Fig. 4.3, with K = 0.0289, IA = 1.10.

From Equation (4.2): ( ) ( )

ρAs

sAsi si

pbv

EI=

−=

× −×

× =1 100 10 1

10

2 2

3

0.3

81.91.10 12.2 mm

From Fig. 4.5(b), with l /b = 1, d/b = 40/10 = 4 (i.e. b/d = 0.25), and νs = 0.3, SA = 0.88.

From Equation (4.4): ρ ρA A AS

si= = × =0.88 12.2 10.8 mm

(b) From Fig. 4.3, with K = 0.0289, IAB = 0.26. Therefore ρABsi

= 2.9 mm

From Fig. 4.5(b), SAB = 1.00. Thus ρρρρAB = 2.9 mm (c) From Fig. 4.4, with K = 0.0289, Msi = 0.005.

From Equation (4.3): m plbM= = × × × =100 10 10 0.005 50 kNm / m

From Fig. 4.7(b), R = 1.00. Thus m = 50 kNm/m.

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4.1 Multi-Layered Soil Profiles The charts given in §4 deal with a raft foundation resting on a single soil layer. As we know, the majority of situations involve multi-layered soil profiles. Fraser and Wardle (1976) suggested the following method for dealing with multi-layered soil profiles. 1. The total settlement, ρ, and the maximum bending moment, m, are calculated by means of

Equations (4.1) and (4.2), using equivalent elastic parameters, Es* and ν , where ( )E E* = −1 2ν .

2. The equivalent elastic parameters are calculated by using the following relationships:

( )

ρ( ) ( )z pbv

EI z

s

=−1 2

(4.6)

where: ρ(z) is the vertical settlement at depth, z, below the centre of a uniformly loaded, square, flexible raft; and

I(z) is shown plotted in Figure 4.12.

1 1

1E E

I

Is i

i

i

n

* *==∑

∆∆ total

(4.7)

ν ν==∑ i

i

i

n I

I

∆∆ total1

(4.8)

3. The task of determining Es

* is simplified by plotting values of 1 Ei* using a horizontal scale

which is linear with respect to I(z), but for convenience is labelled with values of z/b, as shown in Figure 4.13. The value of 1 E * is then simply the average value of 1 Ei

* weighted according to the special horizontal scale.

4. The average Poisson’s ratio, ν , is calculated in the same way, that is, by plotting n against z/b, using the special scale.

Figure 4.12 Vertical settlement influence factor, I(z). (Source: Fraser and Wardle, 1976.)

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Figure 4.13 Variation of 1 Ei* with depth z/b. (Source: Fraser and Wardle, 1976.)

Example 4.2 Using the same situation as in the previous example, except that, in this case, the soil is multi-layered, as shown in Figure 4.14, determine (a) the central settlement, (b) the differential settlement between the centre and the mid-edge of the raft, and (c) the maximum bending moment.

Figure 4.14 Details of Example 4.14. (Source: Fraser and Wardle, 1976.)

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Solution: • Plot 1 E* to special scale. For example:

Layer 1: E1* = 100 MPa, ∴1 1 1001E* = = 0.010 .

Layer starts at z = 0, that is z/b = 0/10 = 0, and Layer ends at z = 10, that is z/b = 10/10 = 1.

Layer 2: E2

* = 80 MPa, ∴1 1 802E* = = 0.0125. Layer starts at z/b = 10/10 = 1, and Layer ends at z = 20, that is z/b = 20/10 = 2.

etc. as shown in Figure 4.13.

• Now calculate the area under the curve:

Area1.12 0.48

100

0.48 0.26

80

0.26 0.18

60

0.18 0.14

1000.01088= − + − + − + − =

• And the average value = 1.12 0.14

0.0108890.0 MPa

− =

• ν = 0.3 . Therefore: ( ) ( )E Es s s= − = − =* 1 90 12 2ν 0.3 81.9 MPa

• Analyses are then performed in the same manner as in Example 4.1. Hence, the results are the same as in Example 4.1. Geotech3_LS8_Shallow Foundations.doc 2006, M. B. Jaksa