Geotechnical Engineering-II [Lec #9+10: Westergaard Theory]

Class notes of Geotechnical Engineering course I used to teach at UET Lahore. Feel free to download the slide show.
Anyone looking to modify these files and use them for their own teaching purposes can contact me directly to get hold of editable version.

1. 1
Geotechnical Engineering–II [CE-321]
BSc Civil Engineering –5
th
Semester
by
Dr. Muhammad Irfan
Assistant Professor
Civil Engg. Dept. –UET Lahore
Email:[email protected]
Lecture Handouts: https://groups.google.com/d/forum/geotech-ii_2015session
Lecture # 9
4-Oct-2017

2. 2
STRESS UNDER UNIFORMLY LOADED
IRREGULAR SHAPED AREA
How to determine stress in soil caused by irregularly shaped
loaded areas?
Newmark (1942)influence charts
Determination of stresses at given depth and location (both
within and outside the loaded area)
Vertical stress
Horizontal stress
Shear stress

3. 3
•Based on Bousinesqtheory
•Similar charts available for
Westergaardtheory (to be
discussed later)
STRESS UNDER UNIFORMLY LOADED IRREGULAR
SHAPED AREA
–Newmark Influence Charts –

4. 4
•Contours of a cone
•Each ‘area’ or ‘block’ has the
same surface areain cross-
section
•Projectiononpaperdistortsthe
blockarea,i.e.areaslook
smallerclosetothecenterand
viceversa
–NEWMARK
INFLUENCE CHARTS –

5. 5
•Drawing to be made on scale
•Distance A-B equal to depth of
interest
•Scale of loaded area to be
selected accordingly
•Center of influence chart to
coincide with point of interest
•Count number of blocks under
loaded area
–NEWMARK
INFLUENCE CHARTS –
∆??????
??????=??????
??????.??????.(????????????.??????????????????�????????????�??????)
q
o= contact stress
I= influence factor

6. 6
Practice Problem #8
Whatistheadditional
verticalstressatadepthof10
munderpointA?
No of elements = 76 (say)
∆??????
??????=??????
??????.??????.(????????????.??????????????????�????????????�??????)
A B
I = 1/200
20mm

7. 7
STRESS
DISTRIBUTION
CHARTS
Pressureisobars(alsocalledpressure
bulbs)basedontheBoussinesq
equationforsquareandstripfootings.
Applicableonlyalonglineabfromthe
centertoedgeofthebase.
Ref:Bowlespp#292
Fig.5-4

8. 9
STRESS INCREASE (∆q) DUE TO
EXTERNAL LOAD
Determination of stress due to external load at any
point in soil
1.Approximate Method
2.Boussinesq’sTheory
3.Westergaard’sTheory

9. 10
Westergaard’sTheory
•Boussinesqtheory derived for homogeneous, isotropic, linearly
elastic half-space.
•Many natural soils sedimentary(layered) in nature; e.g. varved
clays.
•Westergaardtheory considers infinitely thin elastic layers of soil.

10. 11
Westergaard’sTheory for Point Load
Westergaard, proposed (1938) a formula for the computation of vertical
stress s
zby a point load, P, at the surface as;  
  
23
2
2
221
2221
2
zr
z
P
z






s  
23
2
2
21
1
zr
z
P
z



s
If poisson’sratio, , is taken as zero, the above equation simplifies to
Where,  
23
2
21
11
zr
I
W


 WI
z
P
2

Independent of all
material properties.

11. 12
Westergaardvs BoussinesqCoefficient 
25
2
1
1
2
3
zr
I
B


  
23
2
21
11
zr
I
W



The value of I
Wat r/z= 0is
0.32 which is less thanthat of
I
Bby 33%.
Boussinesq’ssolution gives
conservative results at shallow
depth.

12. 13
Westergaard
Charts for
Rectangular
Loads
Influencevaluesforvertical
stressundercornersofa
uniformlyloadedrectangular
areaforWestergaardtheory
(afterDuncan&Buchignani,
1976)
Ref: Holtz & Kovacs (2
nd
Ed.)
Fig. 10.9 (pp #480)

13. 14
Influence values for vertical stress under
center of a square uniformly loaded area
(Poisson’s Ratio, ν= 0.0)
(after Duncan & Buchignani, 1976)
Ref: Holtz & Kovacs (2
nd
Ed.)
Table 10.1 (pp #481)

14. 15
Influence values for vertical stress under
center of infinitely long strip load.
(after Duncan & Buchignani, 1976)
Ref: Holtz & Kovacs (2
nd
Ed.)
Table 10.2 (pp #481)

15. 16
Influence values for vertical stress
under corner of a uniformly loaded
rectangular area.
(after Duncan & Buchignani, 1976)
Ref: Holtz & Kovacs
(2
nd
Ed.)
Table 10.2 (pp #481)

16. 17
SUMMARY
WESTERGAARD METHODBOUSSINESQ METHOD
APPROXIMATE METHOD
Use of 2:1 (V:H) stress distribution
??????
??????=
??????
(??????+??????)∙(??????+??????)Bz I
z
P
2
s  
25
2
1
1
2
3


zr
I
B
 Wz I
z
P
2
s
Where,  
23
2
21
11
zr
I
W



Where,

17. 18
Practice Problem #9

18. 22
CONCLUDED
REFERENCE MATERIAL
An Introduction to Geotechnical Engineering (2
nd
Ed.)
Robert D. Holtz & William D. Kovacs
Chapter #10