Footings - Examples

TYPES OF SHALLOW FOOTINGS

(a) Isolated footing

75mm (min) cover to earth face

150mm (min)

B Bmin = bc + 300 mm B is determined by satisfying serviceability and ultimate limit states.

de - effective depth determined from bending and shear

bc - column width

Critical section for bending

Critical section

for shear

21

(b) Isolated footing and floor

slab

(c) Combined footing (d) Mat foundation

(e) Waffle mat (f) Tendons layout in a post-tensioned mat

(g) Strip footing

bw (min)

2bw

bw - wall width

0.5 bw (min)

(h) A strip footing under construction

(i) Location of footings near slopes

(j) Location of footings on s slope

(k) Stepped footings

Figure 4.1 Shallow footings and approximate initial dimensions.

There are several types of shallow foundations. Some are shown in Figure 4.1. A spread or

individual forting, also called isolated footing, transfers individual column loads directly to the

soil (Figures 4.1a,b). If a single spread footing interferes with another spread footing, the two can

be combined to form a combined footing (Figure 4.1c). When there is a large number of spread

footings required for a structure it may he more economical to use a mat or raft foundation. If the

overburden pressure of a soil removed to embed a raft foundation is equal to the stress imposed

by the structure, the raft is termed fully compensated or floating raft (Figure 4.1d). If the

overburden pressure is a fraction of the imposed stress, the raft is partially compensated. Some

rafts are thickened in the form of a waffle as shown in Figure 4.1e. Post-tensioned mat or floor

slab are often located just below the surface (Figure 4.1f). Raft foundations are also integrated

with pile foundations. These are called pile-raft foundations. Walls are normally supported

using a strip foundation (Figures 4.1g,h). A strip foundation is one in which the width (B) to

length (L) ratio is small (B/L >> 10). On sloping ground, a footing can rest on top of the slope,

on the slope or can be stepped as illustrated in Figures 4.1i,j,k.

THEORETICAL BEARING CAPACITY BASED ON LIMIT EQUILIBRIUM METHOD

The theoretical bearing capacity equations in common use in design practice were

obtained using the limit equilibrium method. Prandtl (1920) showed theoretically that a wedge

of material is trapped below a rigid plate when it is subjected to concentric loads. Terzaghi

(1943) applied Prandtl's theory to a strip footing (length is much longer than its width) with the

assumption that the soil is a semi-infinite, homogenous, isotropic, weightless rigid-plastic

material. Based on Prantls theory, failure of the footing occurs by a wedge of soil below the

footing pushing its way downward into the soil. The failure mechanism postulated by Terzaghi

(1943) is shown in Figure 4.2a.

Df F

Considered by Meyerhof

Df

Df

Figure 4.2 Failure mechanisms

The zone ABC is a rigid wedge of soil trapped beneath the footing. Zone ABD is fan like

with radial slip lines stopping on a logarithm spiral slip line. The other, zone ADE, consists of

slip lines oriented at angles of 2

45 + and 2

45 to the horizontal and vertical planes,

respectively. The zone ADE is called the Rankine passive zone. The surfaces, AB and AD, are

frictional sliding surfaces d their actions are similar to rough walls being pushed into the soil.

The pressure exerted is called passive earth pressure. The shape of the surface BD depends on

the ratiozo

B where is the unit weight of the soil and zo is the overburden pressure. For a

weightless soil, ,zo

B and BD is a logarithmic spiral; for B = 0, BD is a circle; for 0B ,

BD is between a circle and a logarithm spiral. For clean, coarse-grained soils, BD is always a

circle (Vesic, 1973). For dense coarse-grained soils or heavily over-consolidated fine-grained

soils, the failure planes are expected to reach the ground surface and the failure mode is called

general shear failure. or loose coarse-grained soils or lightly over-consolidated fine-grained F an0

soils, the failure planes, if they are developed, are expected to lie within the soil layer below the

base of the footing and extend laterally. The failure mode is called local shear. Another type of

shear failure is possible. For very loose soil, the failure surfaces may be confined to the surfaces

of the rigid wedge. This type of failure is termed punching shear.

Each of these collapse mechanism depends on the relative density of the soil and the depth of

embedment (Figure 4.3).

Figure 4.3 Collapse mechanism based on relative density and embedment depth (Source: Vesic,

1973)

In saturated fine-grained soils, the settlement to mobilize general shear failures under a surface

footing is 3% to 7% of the foundation width and about 15% for deeply embedded footings. For

coarse-grained soils, the corresponding settlements are 5% to 15% for surface footings and 25%

for deeply embedded footings.

We will consider the theoretical bearing capacity equations for short-term (undrained condition)

and long-term (drained condition) separately. A total stress analysis (TSA) utilizing the

undrained shear strength, su, is employed for short-term loading of fine-grained soils. An

effective stress analysis (ESA) utilizing the effective friction angle, , is employed for long-term and short-term loading of fine-grained and coarse-grained soils. All the equations are based on

the following assumptions.

1. The embedment depth is not greater than the width of the footing (Df < B)

2. General shear failure occurs

3. The shear strength of the soil above the footing base is negligible (Terzaghis and Hansens

assumption).

4. The soil above the footing base can be replaced by a surcharge stress (= Df). 5. The soil is a rigid-plastic, homogenous medium. No displacement occurs prior to collapse.

The appropriate friction angle to use for an ESA is the peak friction angle,p. However, p is not a fundamental soil parameter. Its magnitude depends on the ability of the soil to dilate which

in turn depends on the relative density and the normal effective stress.

We will consider the theoretical net ultimate bearing capacity (qult) of shallow footings

based on limit equilibrium proposed by Meyerhof (1963), Vesic (1973) and Hansen (1970).

Terzagis equation is not given because it has been generally superseded in engineering practice

by Meyerhof (1963), Vesic (1973) and Hansen (1970) methods. The author has modified some

of these equations to separate total stress analysis from effective stress analysis.

Meyerhofs (1963) equations

Vertical load

TSA: fccuult Ddss14.5q +=

ESA: ult f q q qq D (N 1)s d 0.5 B N s d = +

Inclined load

TSA: fccuult Dids14.5q +=

ESA: ult f q q q qq D (N 1)d i w 0.5 B N d i w = +

where

ptan p2 oqN e tan (45 )2

= + ; qN (N 1) tan(1.4 p )= and the other factors are given in Table 4.1.

Vesics (1973) equations

Vertical load

TSA: fccuult Ddss14.5q +=

ESA: ult f q q q qq D (N 1)s d w 0.5 B N s d w = +


= + ; qN 2(N 1) tan p= + and the other factors are given in Table 4.1.

Hansens (1970) equations

All loads

TSA: qult = 5.14 su (1 + sc + dc - ic - bc - gc)

ESA: qult = Df (Nq - 1) sq dq iq bq gq wq + 0.5 B N s d b g w where


= + ; qN 1.5(N 1) tan p= and the other factors are given in Table 4.1

The factors, Nq and N, are bearing capacity factors that are functions of p; sc, sq and s are shape factors, dc, dq, d are embedment depth factors, ic, iq, i are load inclination factors, bc, bq,

b are base inclination (base tilt) factors, gc, gq, g are ground inclination factors, wq and w are

ground water factors. Various equations have been proposed for the various factors as given in

Table 1. The bearing capacity equations apply for a single resultant load with normal Vn and

horizontal components HB parallel to the width, B (the short side) and horizontal components HL

parallel to the length, L (the long side). When investigating potential failure along the short side,

use HL and for failure along the long side, use HB.

Table 4.1 Geometric factors for use in theoretical bearing capacity equations

Shape factors

Meyerhof Vesic

Hansen

sc B1 0.2L+

B1L+

B0.2L ; ; vertical loads only 0 =

cBB0.2iL ; for inclined loads, short side

failure

cBL0.2iB ; for inclined loads, long side

failure sq

p

p2 op p

B1 0.1KL

K tan (45 ); 102

+ = + >

p

B1 tanL + p

B1 sinL + ; ; vertical loads

only

0 =

qB pB1 i sinL + ; for inclined

loads, short side failure

qL pL1 i sinB + ; for inclined loads,

long side failure s

p

p2 op p

B1 0.1KL

K tan (45 ); 102

+ = + >

=

B1 0.4L p

B1 0.4L + ; ; vertical loads

only

0 =

B

L

B i1 0.4

L i

; for inclined loads, short side failure, s > 0.6 should be used

L

B

L i1 0.4

B i

; for inclined loads, long side failure, s > 0.6 should be used

Depth factors

Meyerhof Vesic

Hansen

dcfD1 0.2

B+

fD1 0.4B

+ ; fD B ; see note 1

1 fD1 0.4 tanB

+ ; fD B> ; see note 1

fD0.4B ; fD B ; see note 1

1 fD0.4 tanB

; fD B> ; see note 1

dq ofp p

D1 0.1 K ; 10B

+ >

2 fp p

2 -1p p

D1+2 tan (1-sin ) B

1+2 tan (1-sin ) tan

; see note 1

2 fp p

2 -1 fp p

D1+2 tan (1-sin ) ; D BB

D1+2 tan (1-sin ) tan ( )B

; D > B

see note 1

d ofp p

D1 0.1 K ; 10B

+ >fD < B 1; 1; fD < B

Notes: note 1 set all depth factors to 1 if the shear strength of the soil above the base is small

Load inclination factors

Meyerhof Hansen

ic 2o1

90

; load inclined in direction of

footing width

acos 1 1 sin2

+ ; load inclined in

direction of footing length

a = adhesion factor that is usually 32

to 21

for short-term loading

i

n u

H0.5(1 1 )V A s

; see note 2 Q is the applied vertical load with Vn as the normal load, Hi is the horizontal load, the subscript i denotes direction.

B HL

HB

+

L

Vn Q

iq 2o

190


footing width

p

sincos 1sin

; load inclined in


5 o 0ip q

n

H(1 0.5 ) ; 45 30 ;i 0V

> ; see note 3

5o

io 0

n

(0.7 )H4501 ; for

V

90=

i 2o1

90


footing width

p

sincos 1sin

; load inclined in


5 o 0ip

n

H(1 0.7 ) ; 45 30 ;i 0V

> ; see note 2

5o

io 0

n

(0.7 )H4501 ; for

V

90=

Notes: note 2 for the horizontal force along the width, Hc q cB qB Bi , i , i i , i , i =

i = HB, (failure along the long

side, HB dominant) or for the horizontal force along the long side, Hc q cL qL Li , i , i i , i , i = i = HL, (failure along short side, HL dominant)

Base and ground inclination factors (see Figure ) Hansen

bc oo o

p; ; 9147 < + < o0

bq po o o

p

exp( 2 tan ); i s in radians

; 90

< + <

b po o o

p

exp( 2.7 tan ); i s in radians

; 90

< + <

gc oo o

p;; ; 90147 < + < o

gq 5 o op(1 0.5 tan ] ; ; 90 < + < o

g 5 o op(1 0.5 tan ] ; ; 90 < + < o

Vn Df

B/2 B/2

Hi

+

+

Figure 4.4 Footing on a slope: definition of terms in Hansens equation The effective width B and effective length, L, giving an effective area, A = B L must be used

in the theoretical bearing capacity equations. When the location of the resultant load (load

center) is not coincident with the centroid (center of area) of the footing, the footing dimension is

theoretically adjusted to align the load center with the centroid. Some possible cases are shown

in Figure 4.5.

B

HL

HB +

L

B L

Vn

Q

B L

Vn eB

eL

B L

Vn

My

Mx

B = B, L = L A = A = BL

B = B - 2eB, L = L - 2eLA = BL

x YB L

n n

M Me ;eV V

= = B = B - 2eB, L = L - 2eL A = BL

B L

Vn eB

eL

B = B - 2eB, L = L - 2eL A = BL Use both the inclination factors and the effective width in the equations

Figure 4.5 Some possible cases of eccentric loads

You have to make some adjustment to the theoretical bearing capacity equations for ground

water condition. The term Df in the bearing capacity equations for an ESA refers to the vertical stress of the soil above the base of the foundation. The last term B refers to the vertical stress of a soil mass of thickness, B, below the base of the footing. You need to check which one of three

ground water situations is applicable to your project.

Situation 1: Ground water level at a level B below the base of the footing. No correction is

required, i.e., wq = w = 1

Situation 2: Ground water level within a depth B below the base of the footing

(Figure 4.3a). fD z (B D< < + f )

wq = 1, ( )f f

sat

z-D D zw = (1 )B B

+ + B

z

Figure 4.3 Ground water effects below base of footing.

B

Df

B

z

(Df z) Df

(a) Ground water within a depth B below base (b) Ground water within embedment depth

Situation 3: Ground water level within the embedment depth f0 z D (Figure 4.3b)

qf sat f

z zw = (1 )D D

+ and satw =

DESIGN OF SHALLOW FOOTINGS

It is customary for practicing engineers to make an estimate of the footing size and then check

that it satisfies serviceability and ultimate limit states. Typical initial dimensions for isolated and

strip footings are shown in Figure Remember that these dimensions are only initial guidelines.

The procedures to design a shallow rectangle footing using ASD are as follows:

1. Select a footing size

2. Determine settlement using a load factor of 1.

3. Check that settlement is less than or equal to the serviceability limit state. If settlement

exceeds serviceability limit state, increase the footing size and recalculate. If settlement

is much less (> 10% ) than the serviceability limit state, then reduce footing size and

recalculate.

4. Calculate ultimate bearing capacity (qult) using the bearing capacity equations if lab tests

results are available or from empirical equations using field test data. For fine-grained

soils, calculate the ultimate bearing capacity for both short-term (TSA) and long-term

(ESA) conditions. For free draining coarse-grained soils, calculate the ultimate bearing

capacity for long-term (ESA) conditions, which is also applicable to short-term

conditions.

5. Calculate the allowable bearing capacity using ulta fqq DFS

= +

6. If the eccentricities, y xB LM Me ; eP P

= = , where x and y are the axes parallel to the width

and length respectively, are not zero then calculate B6eB

= or L6eL

=

7. Calculate and BB B 2e = LL L 2e =

8. Calculate where P( )a maxP P(1= +) a is the applied load and is the higher value calculated in item 6.

9. Check that ( )a a maxq B L P

The performance or resistance factors to determine the bearing capacity of shallow footings

based on LFRD are shown in Table 7X.A

Table 7 X. A Performance factors for bearing capacity calculations using LRFD

Resistance

Factor, RESA: Coarse-grained and fine grained soils

cs from lab tests p from lab tests SPT CPT Plate load test

0.95*

0.80

0.45

0.55

0.55

TSA: Fine-grained soils

su from lab tests Shear vane CPT Plate load test

0.60

0.60

0.50

0.55

Rock 0.6

* Authors recommendation

The procedures to design a shallow rectangle footing using LRFD are as follows:

1. Select a footing size.

2. Determine settlement using a load factor of 1.

3. Check that settlement is less than or equal to the serviceability limit state. If settlement

exceeds serviceability limit state, increase the footing size and recalculate. If settlement

is much less (> 10% ) than the serviceability limit state, then reduce footing size and

recalculate.

4. Determine the maximum factored load (Pu) by combining loads according to IBC (2000)

or other codes.

5. Calculate ultimate bearing capacity (qult) using the bearing capacity equations if lab tests

results are available or from empirical equations using field test data. For fine-grained

soils, calculate the ultimate bearing capacity for both short-term (TSA) and long-term

(ESA) conditions. For free draining coarse-grained soils, calculate the ultimate bearing

capacity for long-term (ESA) conditions, which is also applicable to short-term

conditions.

6. If the eccentricities, y xB LM Me ; eP P

= = , where x and y are the axes parallel to the width

and length respectively, are not zero then calculate BB B 2e = and LL L 2e =

7. Calculate uuPq

B L= ( if eB = 0, then B = B; if eL = 0, then L = L)

8. Calculate the gross resistance

R = R qult + i Df

where Df is the effective soil pressure above the footing base, i is the dead load factor for earth pressure (usually 0.9).

9. Check that uR qSuggested changes if serviceability and ultimate limit states are not satisfied after first trial.

Case 1: Settlement too large

Actions: (1) Increase size of footing. This will reduce the applied stress.

(2) Consider a mat foundation if isolated footing becomes too large with the potential for

stress overlaps. The mat foundation will spread the load over a larger area than isolated footings

thereby reducing the applied stress.

(3) Consider soil improvement such as dynamic compaction (coarse-grained soils) and

soil reinforcement by, for example, geotextiles (see Chapter 11).

(4) Consider deep foundations.

Case 2: Ultimate bearing capacity too low.

Actions: Same as above

Vertical loadVertical loadDESIGN OF SHALLOW FOOTINGSTable 7 X. A Performance factors for bearing capacity calcul

Footings - Examples

Documents

Transcript of Footings - Examples