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# Geotechnical Engineering-II [Lec #7A: Boussinesq Method]

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 #7A: Boussinesq Method]

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 # 7A 28-Sep-2017
2. 2. 2 STRESS DISTRIBUTION IN SOIL What causes stress in soil? Two principle factors causing stresses in soil. 1. Self weight of soil 2. External loads (Structural loads, external load, etc.) v h 
3. 3. 3 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
4. 4. 4 Q   2522 3 5 3 2 3 2 3 zr zQ L Qz z     The above relationship for z can be re-written as              2522 1 1 2 3 zrz Q z   where    252 1 1 2 3 zr IB    QBI z Q 2  Independent of all material properties Boussinesq’s Theory for Point Load
5. 5. 5 Practice Problem #5 A concentrated load of 1000 kN is applied at the ground surface. Compute the vertical stress (i) at a depth of 4m below the load, (ii) at a distance of 3m at the same depth. (A) Use Boussinesq’s equation (B) Use Westergaard’s equation P              2522 1 1 2 3 zrz Q z  
6. 6. 6 Vertical Stress caused by Line Load x z z z Q/unit length x A y By integrating the point load equation along a line, stress due to a line load (force per unit length) may be found. Lz I z q    2 2 /1 12         zx IL  Where, q is line load in “per unit length”
7. 7. 7 Practical Problem #6 Following figure shows two line loads and a point load acting at the ground surface. Determine the increase in vertical stress at point A, which is located at a depth of 1.5 m. Q = 10000 kN z 2 m A 1.5 m 2 m 3 m q2 = 250 kN/m q1 = 150 kN/m              2522 1 1 2 3 zrz Q z                22 /1 12 zxz q z   Point Load Line Load
8. 8. 8 STRESS INCREASE (∆q) DUE TO EXTERNAL LOAD  Point load  Line Load • But engineering loads typically act on areas and not points or lines. • Bousinesq solution for line load was thus integrated for a finite area Bz I z Q 2  Lz I z q  Uniformly Loaded Circular Area Uniformly Loaded Rectangular Area Trapezoidal, Triangular, etc.
9. 9. 9 z RO STRESS UNDER UNIFORMLY LOADED CIRCULAR AREA Case-A: Vertical stress under the center of circular footing            232 1 1 1 zR q o z Boussinesq equation can be extended to a uniformly loaded circular area to determine vertical stress at any depth. where, q = UDL (load/area) RO = Radius of footing
10. 10. 10 STRESS UNDER UNIFORMLY LOADED CIRCULAR AREA Case-B: Vertical stress at any point in soil z a RO r z ),( nmIq Zz  where, IZ = Shape function/ Influence factor m = z/RO; n=r/RO RO = Radius of footing r = distance of Δσz from center of footing z = depth of Δσz
11. 11. 11 STRESS UNDER UNIFORMLY LOADED CIRCULAR AREA (Foster & Alvin, 1954; U.S. Navy, 1986) Assumptions: Semi-infinite elastic medium with Poisson’s ratio 0.5. (stress in percent of surface contact pressure)
12. 12. 12 A water tank is required to be constructed with a circular foundation having a diameter of 16 m founded at a depth of 2 m below the ground surface. The estimated distributed load on the foundation is 325 kPa. Assuming that the subsoil extends to a great depth and is isotropic and homogeneous. Determine the stress z at points (i) 10 m below NSL; at center of footing (ii) 10 m below NSL; at distance of 8 m from central axis of footing (iii) 18 m below NSL; at center of footing (iv) 18 m below NSL; at distance of 8 m from central axis of footing Neglect the effect of the depth of the foundation on the stresses. Practice Problem #7
13. 13. 13 STRESS UNDER UNIFORMLY LOADED RECTANGULAR AREA Bousinesq equation can be extended for uniformly loaded rectangular area as; ),( nmIq recz  where, IZ = Shape function/ Influence factor m = b/z; n=l/z x z q A y z dx dy
14. 14. 14 STRESS UNDER UNIFORMLY LOADED RECTANGULAR AREA •14 z L n z B m  , Log scale
15. 15. 15 STRESS UNDER UNIFORMLY LOADED RECTANGULAR AREA This methods gives stress at the corner of rectangular area ),( nmIq recz  A B D C Case I E F G A B D C Case II σz due to ABCD = 4 x σz due to EBFG
16. 16. 16 A 20 x 30 ft rectangular footing carrying a uniform load of 6000 lb/ft2 is applied to the ground surface. Required The vertical stress increment due to this uniform load at a depth of 20 ft below the (i) corner, and (ii) center of loaded area. G A BE D C 20 ft 30 ft F Practice Problem #8
17. 17. 17 E FH G E FH I STRESS UNDER UNIFORMLY LOADED RECTANGULAR AREA G A B D C A B D C I A B D C Case I Case II σz due to ABCD = 4 x σz due to EBFG Case III σz due to ABCD = σz due to (EBFI + IFCG + IGDH + AEIH)
18. 18. 18 E I STRESS UNDER UNIFORMLY LOADED RECTANGULAR AREA A B D C EF GH F A B D C Case IV σz due to ABCD = 2 x σz due to ABEF Case V σz due to ABCD = 2 x σz due to EBCF Case VI σz due to ABCD = σz due to (AEGI – BEGH – DFGI + CFGH) A BE D CF
19. 19. 19 CONCLUDED REFERENCE MATERIAL An Introduction to Geotechnical Engineering (2nd Ed.) Robert D. Holtz & William D. Kovacs Chapter #10