study guide for an exam (geotechnical)
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6StripandSpreadFoundations3.pptGeotechnical Engineering
6
Strip and Spread Foundations
Contents:
Introduction
1.1 Types of foundations
1.2 Footing depth
1.3 Settlement
1.4 Bearing capacity
Vertical stress distribution
2.1 Simple spread assumption
2.2 Bousinesq’s ‘surface point load’
2.3 Fadum’s ‘uniformly distributed rectangular load’
2.4 Newmark’s chart(s)
2.5 Uniform strip load
2.6 Bulb of pressure
1. Introduction
- Foundation provides a critical interface between a structure and the ground beneath it – structural integrity of both sides must be considered
- Three main types: strip, pad & raft
- Three main design criteria
- footing (foundation) depth
- settlement
- bearing capacity
1.1 Types of foundations
1.2 Design criteria:
1.2.1 Footing depth
At a depth as to prevent failure due to changes in surface conditions:
- Water flow – removal of fine soil
- Frost heave – (uniform fine sands and silts, etc) 450mm frost => 600mm foundation depth
- Shrinkage and swelling – action of tree roots
New foundation outside the zone of influence
1.2.2 Settlement (total and differential)
- Total settlement
- Amount and rate is important (small settlement rapidly = more damaging than larger movements over a longer time period)
- Differential settlement
- Relative settlement between different parts of a structural
Total settlement = no problem
Differential settlement = problem
1.2.3 Bearing capacity
- Bearing capacity is the average contact pressure between the footing and the soil which will produce shear failure in the soil divided by FOS (>3)
plastic
elastic
Steel
Soil
2. Vertical stress distribution
Assume that the soil is:
- Homogeneous
- Elastic (i.e. stress proportional to strain)
- Elastically isotropic (Ex=Ey=Ez)
Basic methods:
- Simple spread assumption
- Boussinesq’s ‘surface point load’ solution
- Fadum’s ‘uniformly distribution rectangular load’ solution
- Newmark’s chart(s)
- Uniform strip load solution
2.1 Simple spread assumption
Large loaded areas on thin beds of soil e.g. raft foundations, embankments, etc
Example 1: Simple spread assumption
Estimate the average vertical stress at 5 m depth below a 2 m square foundation, load 200 kN/m2.
200 kN/m2 x 2 m x 2 m = 800 kN
Stress on foundation x Area of foundation = load on foundation
800 kN / (7 m x 7 m) = 16.3 kN/m2
Example 2: Simple spread assumption
Estimate the average vertical stress at 5 m depth below a strip foundation 1.5 m, load 90 kN/m2.
90 kN/m2 x 1.5 m = 135 kN/m
Stress on foundation x width of foundation = load on foundation per unit length
135 kN/m / 6.5 m = 21 kN/m2
2.2 Bousinesq ‘surface point load’ solution
- Single point load / Semi-infinite / Elastic Medium
- Polar co-ordinates
for vertical stress
where Iz (influence factor)
Iz can be plotted against r/z
Example 3a: 1m square foundation point load 50kN, Calculate
(a) stress 4m directly beneath the centre
- Points below the load
r = 0
and at all depths
r/z = 0
from graph Iz vs r/z
σ = 0.4775 x 50 kN = 1.49 kN/m2
42 m2
Example 3: A 1m square foundation point load 50 kN, Calculate
(b) a point 3m below the foundation offset by a distance of 5m
b) Points below the load
r = 5 m
and at
r/z = 5/3 = 1.667
from graph Iz vs r/z
s = 0.017 x 50 kN = 0.094 kN/m2
32 m2
0.017
Bousinesq’s method is only for a point load, however it can be extended by the principle of superposition to cover the case of a foundation exerting a uniform pressure:
Example 4: Using the theory of superposition find the stress at a depth of 3 m beneath the corner of a 3 m square raft which carries a UDL of 200 kN/m2
Divide area into number of equal sections
Total vertical stress at that point = sum of all the stresses
Calculate the load due to stress acting on that section
200 kN/m2 x 1.5 m x 1.5 m (area of small section) = 450 kN
Assume the load acts at the centre of each section
Calculate the vertical stress at the point in question for each section
A
r at point ‘A’
B
r at point ‘B’
1.06 x 3 = 3.18 m
C
D
r at point ‘C & D’
1.5 m + 0.75 m = 2.25 m
0.75 m
At Point A r = 1.06m
At Point B r = 3.18m
At Point C & D r = 2.37m
s = 0.356 x450 kN = 17.8 kN/m2
32 m2
s = 0.073 x 450 kN = 3.65 kN/m2
32 m2
s = 0.142 x 450 kN = 7.1 kN/m2
32 m2
Total stress = 17.8+3.65+7.1+7.1
= 35.7 kN/m2
2.3 Fadum‘Uniformly distributed rectangular load’ solution
Fadum (1948) developed influence factors for the distribution of vertical stress under the corner of a uniformly loaded rectangular foundation length = L, Breath =B
Using above method and superposition it is possible to calculate the vertical stress beneath a wide range of foundation shapes
Example 5: A 4 m x 3 m rectangular foundation, UDL of 100 kN/m2.
Calculate the vertical stress
(a) 2 m beneath the corner of the foundation
0.22
=22 kN/m2
Example 5: A 4 m x 3 m rectangular foundation, UDL of 100 kN/m2.
Calculate the vertical stress
(b) 2 m beneath the centre of the foundation
0.152
=60.8 kN/m2
Example 6: UDL of 100 kN/m2, Calculate the vertical stress (a) Z and (b) Z1 at a depth of 4 m
=31.7 kN/m2
Area 1
Area 2
Area 3
Iz=0.06
Iz=0.132
Iz=0.125
1 m
3 m
3 m
Z1
1 m
2 m
Iz=0.178
Iz=0.075
Iz=0.072
Iz=0.06
subtract
subtract
plus
equals
=11.5 kN/m2
6(c) Two foundations ‘A’ and ‘B’ each 2.5 m square are situated 4 m apart, as shown below, the load on the centre of each foundation is 1000 kN.
(i) Using Boussinesq’s point load solution calculate the increase in vertical stress at a depth of 5 m under the centre of footing ‘A’ due to its load and the load carried by footing ‘B’.
(ii) Using Fadum’s graphical solution calculate the increase in vertical stress at a depth of 5 m under the corner marked X on footing ‘A’ due to its load and the load carried by footing ‘B’.
Boussinesq’s point load solution
or
= 0.4775
For point A r = 0
= 0.04
or
Iz=0.04
= 20.7 kN/m2
r
Equivalent UDL =
160 kN/m2
Fadum’s graphical solution
Stress at 5 m beneath ‘X’ from foundation ‘A’
Stress at 5 m beneath ‘X’ from
foundation ‘B’ & area ‘C’
Stress at 5 m beneath ‘X’ for area ‘C’
C
0.085
0.126
0.11
=16.2 kN/m2
2.4 Newmark’s Chart(s)
Newmark (1942) developed a method which can cater for all foundation shapes – ‘influence diagram’
Vertical stress at the origin of the chart
where N=number of blocks
P=load per unit area
- The influence of each of the small areas given on the chart are the same
- Loaded area drawn onto the chart at correct scale
- Iz = 0.001 for each block within the foundation area
- Scale the foundation should be drawn is such that the distance A – B on the chart is equivalent to the depth z at which the stress is to be determined
Example 7: UDL of 100 kN/m2, Calculate the vertical stress Z and Z1 at a depth of 4 m
=29.5 kN/m2
For Z1
Number of blocks approx = 120
=12 kN/m2
1 m
3 m
3 m
Z1
1 m
2 m
2.5 Uniform strip load solution
Vertical stress at any point beneath a strip foundation can be calculated from:
for simplicity the above expression can be re-written as:
Standard values of Is are given in the following table
Standard values of Is have been calculated for varying values of x and z
Example 8: A 1 m wide strip foundation is to support a UDL of 90 kN/m2. Estimate the vertical stress 4 m beneath
a) the centre of the strip
=14.2 kN/m²
Example 8: A 1m wide strip foundation is to support a UDL of 90 kN/m2. Estimate the vertical stress 4 m beneath
b) the edge of the strip
=13.8 kN/m²
Example 8: A 1 m wide strip foundation is to support a UDL of 90 kN/m2. Estimate the vertical stress 4 m beneath
c) 1 m outside the edge of the strip
=11.0 kN/m²
2.6 Bulb of Pressure
- Use contours to view the distribution of stress ‘isobars’ or ‘pressure bulb’
- The size of pressure bulb is proportional to the size of the loaded area – settlement of foundation
Total se)lement = no problem
Differen3al se)lement = problem
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I
x/b z/b 0 0.4 0.8 1 1.25 1.5 2 3 5 10 0.0 1.000 1.000 1.000 0.500 0.000 0.000 0.000 0.000 0.000 0.000 0.2 0.997 0.992 0.909 0.500 0.059 0.011 0.002 0.000 0.000 0.000 0.4 0.977 0.955 0.773 0.498 0.178 0.059 0.011 0.001 0.000 0.000 0.6 0.937 0.896 0.691 0.495 0.258 0.120 0.030 0.004 0.000 0.000 0.8 0.881 0.829 0.638 0.489 0.305 0.173 0.056 0.010 0.001 0.000 1.0 0.818 0.766 0.598 0.480 0.332 0.214 0.084 0.017 0.002 0.000 1.2 0.755 0.707 0.564 0.468 0.347 0.243 0.111 0.026 0.004 0.000 1.4 0.696 0.653 0.534 0.455 0.354 0.263 0.135 0.037 0.005 0.000 1.6 0.642 0.605 0.566 0.440 0.356 0.276 0.155 0.048 0.008 0.000 1.8 0.593 0.563 0.497 0.425 0.353 0.284 0.172 0.060 0.010 0.001 2.0 0.550 0.524 0.455 0.409 0.348 0.288 0.185 0.071 0.013 0.000 2.5 0.462 0.445 0.400 0.370 0.328 0.285 0.205 0.095 0.022 0.002 3.0 0.396 0.385 0.355 0.334 0.305 0.274 0.211 0.114 0.032 0.003 3.5 0.345 0.338 0.317 0.302 0.281 0.258 0.210 0.127 0.042 0.004 4.0 0.306 0.301 0.285 0.275 0.259 0.242 0.205 0.134 0.051 0.006 5.0 0.248 0.245 0.237 0.231 0.222 0.212 0.188 0.139 0.065 0.010 6.0 0.208 0.207 0.202 0.198 0.192 0.186 0.171 0.136 0.075 0.015 8.0 0.158 0.157 0.155 0.153 0.150 0.147 0.140 0.122 0.083 0.025 10 0.126 0.126 0.125 0.124 0.123 0.121 0.117 0.107 0.082 0.032 15 0.085 0.085 0.084 0.084 0.083 0.083 0.087 0.078 0.069 0.041 20 0.064 0.064 0.063 0.063 0.063 0.063 0.062 0.061 0.056 0.041 50 0.025 100 0.013
x/b
z/b00.40.811.251.523510
0.01.0001.0001.0000.5000.0000.0000.0000.0000.0000.000
0.20.9970.9920.9090.5000.0590.0110.0020.0000.0000.000
0.40.9770.9550.7730.4980.1780.0590.0110.0010.0000.000
0.60.9370.8960.6910.4950.2580.1200.0300.0040.0000.000
0.80.8810.8290.6380.4890.3050.1730.0560.0100.0010.000
1.00.8180.7660.5980.4800.3320.2140.0840.0170.0020.000
1.20.7550.7070.5640.4680.3470.2430.1110.0260.0040.000
1.40.6960.6530.5340.4550.3540.2630.1350.0370.0050.000
1.60.6420.6050.5660.4400.3560.2760.1550.0480.0080.000
1.80.5930.5630.4970.4250.3530.2840.1720.0600.0100.001
2.00.5500.5240.4550.4090.3480.2880.1850.0710.0130.000
2.50.4620.4450.4000.3700.3280.2850.2050.0950.0220.002
3.00.3960.3850.3550.3340.3050.2740.2110.1140.0320.003
3.50.3450.3380.3170.3020.2810.2580.2100.1270.0420.004
4.00.3060.3010.2850.2750.2590.2420.2050.1340.0510.006
5.00.2480.2450.2370.2310.2220.2120.1880.1390.0650.010
6.00.2080.2070.2020.1980.1920.1860.1710.1360.0750.015
8.00.1580.1570.1550.1530.1500.1470.1400.1220.0830.025
100.1260.1260.1250.1240.1230.1210.1170.1070.0820.032
150.0850.0850.0840.0840.0830.0830.0870.0780.0690.041
200.0640.0640.0630.0630.0630.0630.0620.0610.0560.041
500.025
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