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 – 5th Semester
by
Dr. Muhammad Irfan
Assistant Professor
Civil Engg. Dept. – UET Lahore
Email: mirfan1@msn.com
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 Bousinesq theory
• Similar charts available for
Westergaard theory (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 area in cross-
section
• Projection on paper distorts the
block area, i.e. areas look
smaller close to the center and
vice versa
– 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 –
∆𝜎𝑧= 𝑞 𝑜. 𝐼. (𝑁𝑜. 𝑜𝑓 𝐵𝑙𝑜𝑐𝑘𝑠)
qo = contact stress
I = influence factor

6. 6
Practice Problem #8
What is the additional
vertical stress at a depth of 10
m under point A?
No of elements = 76 (say)
∆𝜎𝑧= 𝑞 𝑜. 𝐼. (𝑁𝑜. 𝑜𝑓 𝐵𝑙𝑜𝑐𝑘𝑠)
A B
I = 1/200
20mm

7. 7
STRESS
DISTRIBUTION
CHARTS
Pressure isobars (also called pressure
bulbs) based on the Boussinesq
equation for square and strip footings.
Applicable only along line ab from the
center to edge of the base.
Ref: Bowles pp #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’s Theory
3. Westergaard’s Theory

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

10. 11
Westergaard’s Theory for Point Load
Westergaard, proposed (1938) a formula for the computation of vertical
stress sz by a point load, P, at the surface as;
   
       2322
221
2221
2 zrz
P
z






s
   2322
21
1
zrz
P
z



s
If poisson’s ratio, , is taken as zero, the above equation simplifies to
Where,
   232
21
11
zr
IW



WI
z
P
2

Independent of all
material properties.

11. 12
Westergaard vs Boussinesq Coefficient
   252
1
1
2
3
zr
IB



   232
21
11
zr
IW



The value of IW at r/z = 0 is
0.32 which is less than that of
IB by 33%.
Boussinesq’s solution gives
conservative results at shallow
depth.

12. 13
Westergaard
Charts for
Rectangular
Loads
Influence values for vertical
stress under corners of a
uniformly loaded rectangular
area for Westergaard theory
(after Duncan & Buchignani,
1976)
Ref: Holtz & Kovacs (2nd 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 (2nd 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 (2nd 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
(2nd Ed.)
Table 10.2 (pp #481)

16. 17
SUMMARY
WESTERGAARD METHODBOUSSINESQ METHOD
APPROXIMATE METHOD
Use of 2:1 (V:H) stress distribution
𝜎 𝑧 =
𝑄
(𝐵 + 𝑧) ∙ (𝐿 + 𝑧)
Bz I
z
P
2
s
   252
1
1
2
3


zr
IB

Wz I
z
P
2
s
Where,
   232
21
11
zr
IW



Where,

17. 18
Practice Problem #9

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