FOUNDATION - Lowest part of a structure - To transfer load of structure to the soil - Properly designed foundation transfers the load throughout the soil

without overstressing it

Types of foundation 1) Shallow Foundation

- Transmits load to near surface soils - Depth of embedment-to-width ratio of approximately less than 2.5

(Df/B < 2.5) - 0.5 – 2m deep

a) Spread footing

- Enlargement of a load-bearing wall or column that makes it possible to spread the load of the structure over a larger area of the soil

b) Mat foundation

- Concrete pad where entire structure is constructed - Used in soil with low load-bearing capacity in which the size of

the spread footing required is impractically large - Used in soil with uneven bearing capacity to allow uniform

settlement 2) Deep Foundation

- Transmits most or all of the structural loads to deeper soils - Depth of embedment-to-width ratio greater than 2.5 (Df/B > 2.5)

a) Pile Foundation

Friction Piles – structural load is resisted by the shear stresses generated along

the surface of the pile End Bearing Piles - the load carried by the pile is transmitted at its tip to a firm

stratum of soil

b) Drilled Shaft Foundation - a shaft is drilled into the subsoil and is then filled with concrete

- diameter of drilled shaft is generally larger than that of a pile

Figure 8. Common types of foundations:

a) Spread Footing

b) Mat Foundation

c) Pile Foundation

d) Drilled Shaft Foundation

SOIL BEARING CAPACITY FOR SHALLOW FOUNDATIONS

Bearing Capacity

- Ability of soil to support a load bearing on the surface of the soil or load

(caused by building, bridge, storage tank, or soil embankment)

embedded at depth below the surface

The ff. must be considered in designing foundation:

1) stability of foundation from strength and bearing capacity of soil

2) settlement of foundation is within tolerable limits

Note:

1) Buildings (factories, office blocks, homes, schools, hospitals)

- foundation design is often governed by very low settlement

tolerances than bearing capacity issue

- foundations are dimensioned to give low applied pressures to limit

settlement or piles may be used

2) Large storage tanks

- bearing capacity is the dominant consideration in the design since

storage tanks are typically made of steel that enables them to tolerate

large settlements especially between their perimeter wall and center

of steel floor

- often located close to port facilities that frequently consist soft

normally consolidated clays with low shear strength and high

compressibility

Ultimate Bearing Capacity for Shallow Foundation - failure load of soil - load per unit area of the foundation at which shear failure in soil occurs - maximum load the foundation can support - strength remaining when a material has been loaded and deformed

beyond its peak strength - in estimating ultimate bearing capacity, it is assumed that soil is on the

point of failure, hence, shear strength is its peak or failure value (not ultimate value)

Consider the case of a long rectangular footing of width, B located at the surface of a dense sand layer (stiff soil). - When a uniformly distributed load of q per unit area is applied to the

footing, it settles. - When the value of q is increased, the settlement of the footing gradually

increases. - When the value of q=qu is reached, bearing capacity failure occurs;

footing undergoes a very large settlement w/o any further increase of q - The soil on one or both sides of the foundation bulges, and the slip

surface extends to the ground surface

Two Models of Bearing Capacity Failure 1) General Shear Failure

Consider the case of a long rectangular footing of width, B located at the

surface of a dense sand layer

Figure 9. General Shear Failure of Soil

- Common failure mode in dense sand with relative density greater than

70% (well-defined slip planes)

- Settling of foundation occurs upon application of the load, and a

triangular wedge-shape zone I is pushed downward

- These in turn presses zones II and III sideways and upward

Zone II – radial shear zone (shape of shear planes to be logarithmic

spirals)

Zone III – linear shear zone (soil shears along planar surfaces)

- Bulging on the ground surface adjacent to foundation - At ultimate load qu,(ultimate bearing capacity), surface passes into a

state of plastic equilibrium and failure occurs by sliding

Features: Soil on both sides bulges out; slip surface extends to the

ground surface.

2) Local Shear Failure of Soil

Consider the case of a long rectangular footing of width B located at the

surface of a medium dense sand layer

Figure 10. Local Shear Failure of Soil

- Common in loose sand (not well-defined slip planes) - Significant settlement upon loading - Settling of foundation pushes triangular wedge-shape zone I of soil

downwards, slip surfaces do not extend to the ground surface

- Failure surface first develops right below the foundation and then slowly extends outwards with load increments

- Foundation movement shows sudden jerks first and then after a considerable amount of movement the slip surface may reach the ground.

- A small amount of bulging may occur next to the foundation

Features: Outward bulging of soil is evident; slip surface does not

extend to the ground surface

TERZAGHI’S ULTIMATE BEARING CAPACITY EQUATION

Terzaghi (1943) developed a general expression for bearing capacity of soil

attributed by three components:

1) cohesive strength of soils (c Nc)

2) depth of soil above the level of foundation base (q Nq)

3) self-weight of soil below the level of base of the foundation (½

γBNγ)

Hence, the general expression for bearing capacity of soil is given as:

𝒒𝒖 = 𝒄𝑵𝒄 + 𝒒𝑵𝒒 +𝟏

𝟐𝜸𝑩𝑵𝜸

where:

qu = ultimate bearing capacity

c = cohesion

q = surcharge

= overburden pressure

= vertical effective stress at the level of the foundation

γ = unit weight of soil

B = width of footing

Nc, Nq, Nγ = bearing capacity factors

Nc = cohesion factor Nq = surcharge factor Nγ = self-weight factor

Figure 11. Shallow strip footing used by Terzaghi for the bearing

capacity analysis of foundation

Figure 12. Terzaghi’s bearing capacity analysis

Assumptions:

- No soil consolidation occurs

- Foundation is very rigid relative to the soil

- Soil above the foundation has no shear strength. Only a surcharge load

against the overturning load

- Applied load is compressive and applied vertically to the centroid of the

foundation

- No applied moments present

- Shallow foundation criterion: depth Df ≤ width B

- Weight of the soil above the base of the footing may be replaced by an

equivalent surcharge, q=γDf

MODIFIED TERZAGHI’S ULTIMATE BEARING CAPACITY EQUATIONS

Failure mechanism is different based on shape of footing and alters the

value of ultimate bearing capacity. Hence, Terzaghi and Peck (1967) applied

shape factors resulting to the ff. equations below. These are widely used

and believed to be conservative:

General Shear Failure Mode

1) Square Footing: qu = 1.3 cNc + qNq + 0.4 γBNγ

2) Circular Footing: qu = 1.3 cNc + qNq + 0.3 γBNγ

3) Strip Footing: qu = cNc + qNq + ½ γBNγ

Local Shear Failure Mode

𝑐′ = 2

3𝑐 tan 𝜙′ =

2

3𝑡𝑎𝑛𝜙

1) Square Footing: qu = 1.3 c’N’c + qN’q + 0.4 γBN’γ

2) Circular Footing: qu = 1.3 c’N’c + qN’q + 0.3 γBN’γ

3) Strip Footing: qu = c’N’c + qN’q + ½ γBN’γ

EFFECT OF GROUNDWATER TABLE

In developing the bearing capacity equations, it is assumed that the

groundwater table (GWT) is located at a depth much greater that the width

B of the footing.

Case I: Groundwater table is located at a distance D above the bottom of

the foundation

𝒒 = 𝜸(𝑫𝒇 − 𝑫) + 𝜸′𝑫

𝜸 = 𝜸′

Figure 13. Diagram for Case I

where:

q = equivalent surcharge

= vertical effective stress at the level of the foundation

γ = unit weight of soil

γ’ = effective unit weight of soil

Case II: Groundwater table coincides with the bottom of the foundation

𝒒 = 𝜸𝑫𝒇

𝜸 = 𝜸′

Figure 14. Diagram for Case II

Case III: Groundwater table is at a depth D below the bottom of the

foundation

𝒒 = 𝜸𝑫𝒇

for D ≤ B:

𝜸 = 𝜸𝒂𝒗𝒆 = 𝟏

𝑩[ 𝜸𝑫𝒇 + 𝜸′(𝑩 − 𝑫)]

for D > B:

𝜸 = 𝜸

Figure 14. Diagram for Case III

FACTOR OF SAFETY

𝑞𝑎𝑙𝑙𝑜𝑤 = 𝑞𝑢

𝐹𝑆

FS is a function of:

1) soil type 2) extent of site characterization 3) soil variability 4) structure type

- usually 2.5 – 3.0 (to minimize settlements) while 3-5 (to calculate

allowable bearing capacity)

- Tolerable settlement of foundations for safety of structure at ultimate

load:

Sandy soils: 5-25% of footing width

Clayey soils: 3-15% of footing width

- With deep foundations, weight of soil itself becomes a component of

bearing capacity and has implications with respect to how the safety

factor should be defined and applied

ALLOWABLE BEARING CAPACITY OF SHALLOW FOUNDATIONS

- Design bearing capacity

- Where factor of safety is applied to ultimate value

a) Gross allowable bearing capacity

𝑞𝑔𝑟𝑜𝑠𝑠 𝑎𝑙𝑙𝑜𝑤 = 𝑞𝑢

𝐹𝑆

Where

qu = ultimate bearing capacity

FS = factor of safety

A = area of foundation

b) Net allowable bearing capacity

𝑞𝑛𝑒𝑡 𝑎𝑙𝑙𝑜𝑤 = 𝑞𝑢 𝑛𝑒𝑡

𝐹𝑆=

𝑞𝑢 − 𝑞

𝐹𝑆

Where:

q = γDf = vertical effective stress at the level of the foundation

qu net = ultimate net bearing capacity

= maximum pressure the soil can support above its current

overburden pressure

c) Gross allowable bearing capacity with a factor of safety with respect

to shear failure

𝑞𝑔𝑟𝑜𝑠𝑠 𝑎𝑙𝑙𝑜𝑤 = 𝑐𝑑 𝑁𝑐 + 𝑞 𝑁𝑞 + 1

2𝛾 𝐵 𝑁𝛾

Where:

Nc, Nq, Nγ = bearing capacity factors for friction angle φd

φd = developed angle of friction of soil

tan φd = 𝑡𝑎𝑛𝜙

𝐹𝑆

cd = developed cohesion

= c/FS

FS = Factor of Safety

Table 4. Terzaghi’s Bearing Capacity Factors – NC, Nq, Nγ – (General Shear

Failure)

Φ (deg) NC Nq Nγ Φ (deg) NC Nq Nγ

0 5.70 1.00 0.00 26 27.09 14.21 9.84

1 6.00 1.10 0.01 27 29.24 15.90 11.60

2 6.30 1.22 0.04 28 31.61 17.81 13.70

3 6.62 1.35 0.06 29 34.24 19.98 16.18

4 6.97 1.49 0.10 30 37.16 22.46 19.13

5 7.34 1.64 0.14 31 40.41 25.28 22.65

6 7.73 1.81 0.20 32 44.04 28.52 26.87

7 8.15 2.00 0.27 33 48.09 32.23 31.94

8 8.60 2.21 0.35 34 52.64 36.50 38.04

9 9.09 2.44 0.44 35 57.75 41.44 45.41

10 9.61 2.69 0.56 36 63.53 47.16 54.36

11 10.16 2.98 0.69 37 70.01 53.80 65.27

12 10.76 3.29 0.85 38 77.50 61.55 78.61

13 11.41 3.63 1.04 39 85.97 70.61 95.03

14 12.11 4.02 1.26 40 95.66 81.27 115.31

15 12.86 4.45 1.52 41 106.81 93.85 140.51

16 13.68 4.92 1.82 42 119.67 108.75 171.99

17 14.60 5.45 2.18 43 134.58 126.50 211.56

18 15.12 6.04 2.59 44 151.95 147.74 261.60

19 16.56 6.70 3.07 45 172.78 173.28 325.34

20 17.69 7.44 3.64 46 196.22 204.19 407.11

21 18.92 8.26 4.31 47 224.55 241.80 512.84

22 20.27 9.19 5.09 48 258.28 287.85 650.67

23 21.75 10.23 6.00 49 298.71 344.63 831.99

24 23.36 11.40 7.08 50 347.50 415.14 1072.80

25 25.13 12.72 8.34

Table 5. Terzaghi’s Bearing Capacity Factors – N’C, N’q, N’γ – (Local Shear

Failure)

Φ (deg) NC Nq Nγ Φ (deg) NC Nq Nγ

0 5.70 1.00 0.00 26 15.53 6.05 2.59

1 5.90 1.07 0.005 27 16.30 6.54 2.88

2 6.10 1.14 0.02 28 17.13 7.07 3.29

3 6.30 1.22 0.04 29 18.03 7.66 3.76

4 6.51 1.30 0.055 30 18.99 8.31 4.39

5 6.74 1.39 0.074 31 20.03 9.03 4.83

6 6.97 1.49 0.10 32 21.16 9.82 5.51

7 7.22 1.59 0.128 33 22.39 10.69 6.32

8 7.47 1.70 0.16 34 23.72 11.67 7.22

9 7.74 1.82 0.20 35 25.18 12.75 8.35

10 8.02 1.94 0.24 36 26.77 13.97 9.41

11 8.32 2.08 0.30 37 28.51 15.32 10.90

12 8.63 2.22 0.35 38 30.43 16.85 12.75

13 8.96 2.38 0.42 39 32.53 18.56 14.71

14 9.31 2.55 0.48 40 34.87 20.50 17.22

15 9.67 2.73 0.57 41 37.45 22.70 19.75

16 10.06 2.92 0.67 42 40.33 25.21 22.50

17 10.47 3.13 0.76 43 43.54 28.06 26.25

18 10.90 3.36 0.88 44 47.13 31.34 30.40

19 11.36 3.61 1.03 45 51.17 35.11 36.00

20 11.85 3.88 1.12 46 55.73 39.48 41.70

21 12.37 4.17 1.35 47 60.91 44.54 49.30

22 12.92 4.48 1.55 48 66.80 50.46 59.25

23 13.51 4.82 1.75 49 73.55 57.41 71.45

24 14.14 5.20 1.97 50 81.31 65.60 85.75

25 14.80 5.60 2.25

Sample Problems

1) A continuous footing with cohesion = 19.15 Kpa is shown in the figure.

Use Terzaghi’s bearing capacity factors considering general shear failure

to determine:

a) Gross allowable load per unit area the footing can carry

b) Net allowable bearing capacity with factor of safety equal to four

c) Gross allowable bearing capacity with a factor of safety equal to

four with respect to shear failure

2) A square footing carries an allowable load of 59,130 kg including its own

weight. The bottom of the footing is 1.0 m below the ground surface

and the water table coincide with the bottom of the footing. Assume

general shear failure.

a) Effective surcharge at the bottom of the footing

b) Size of the footing using factor of safety equal to three

c) Allowable bearing capacity

3) A circular footing having a diameter of 1.2 m has its bottom at a depth

of 2.7 m from the ground surface. The water table is located at a depth

of 1.3 m below the ground. The soil has the following properties:

γd = 18.10 KN/m3 c = 15.74 KPa

γsat = 19.30 KN/m3 φ = 20o

a) Assuming local shear failure, find the ultimate bearing pressure

using Terzaghi’s equation.

b) Determine the allowable bearing capacity using a factor of safety of

3.

c) Find the allowable load on the footing.

Practice Problem

A strip footing is to be placed 2m below the surface or soil having a

cohesion of 40 KPa, unit weight of 18.2 KN/m3, and angle of friction of 10o

NC = 9.61 NC’ = 8.02

Nq = 2.69 Nq’ = 1.94

Nγ = 0.56 Nγ’ = 0.24

a) Assuming local shear failure, compute the ultimate bearing capacity of

the footing if the width is 1.25 m. Ans. 287.21 KPa

b) Considering a rectangular footing of 1.25m x 6m and a load factor of

2.5, determine the allowable bearing capacity under general shear

failure.

𝑞𝑢𝑙𝑡 = 𝑐𝑁𝑐 (1 + 0.3𝐵

𝐿) + 𝛾𝐷𝑓𝑁𝑞 + 0.5𝛾𝐵𝑁𝛾(1 − 0.2

𝐵

𝐿)

Ans. 204.97 KPa

c) Find the allowable load that the rectangular footing could carry

Ans. 1537.28 KN