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# 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.

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

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 17. 18 Practice Problem #9
18. 18. 22 CONCLUDED REFERENCE MATERIAL An Introduction to Geotechnical Engineering (2nd Ed.) Robert D. Holtz & William D. Kovacs Chapter #10