14.330 SOIL MECHANICS Assignment #5: Stresses in a Soil Mass.faculty.uml.edu/.../14.3302015Assignment5Solution.pdf · Assignment #5: Stresses in a Soil Mass. PROBLEM #1 (40 Points):

Geotechnical Engineering Research Laboratory Edward L. Hajduk, D.Eng, PE

One University Avenue Lecturer

Lowell, Massachusetts 01854 PA105D

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e-mail: [email protected]

web site: http://faculty.uml.edu/ehajduk

DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING

14.330 SOIL MECHANICS Assignment #5: Stresses in a Soil Mass.

PROBLEM #1 (40 Points): GIVEN: 14.330 Project Site #2 B-1 (see page 3 of assignment handout) and Table 1.

Table 1. Summary of Soil Unit Weights from Boring B-2.

UCSC Symbol γγγγ (pcf) γγγγsat (pcf)

SP (Upper) 106 110

ML 100 107

CH 107 112

SM 112 115

SP (Lower) 114 118

CL 116 120

NOTE: Assume the asphalt and base course layers are removed and replaced with material identical to that underneath them.

REQUIRED: Determine the total, pore pressure, and effective stresses in the soils from the ground surface to bottom of the borehole (i.e. a depth of 75 ft). Provide a plot of these stresses with depth. SOLUTION:

σσσσ’ = σσσσ – u (Effective Stress = Total Stress – Pore Pressure)

σσσσ = γγγγ * Soil Height

u = γγγγw * Height of Water See Figure A for solution. Note values rounded to nearest 5 psf.




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Figure A. Total Stress, Pore Pressure, and Effective Stress with Depth (Static

Conditions). PROBLEM #2 (10 Points): GIVEN: i = 0.05 upward at 14.330 Project Site #2 B-1 (see page 3 of assignment handout) and Table 1. REQUIRED: Plot the effective stresses with depth. What is the effect of the upward water seepage on the effective stress compared to the static conditions?




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SOLUTION:

Upward flow causes an increase in porewater pressure by izγγγγw, where i =

hydraulic gradient (0.05), z = depth below water table, γγγγw = unit weight of water.

Therefore, σσσσ'flow = σσσσ'static - izγγγγw (increase in pore pressure = decrease in effective stress) See Figure B for solution.

Figure B. Total Stress, Pore Pressure, and Effective Stress with Depth (Upward Flow).




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PROBLEM #3 (10 Points): GIVEN: i = 0.05 downward at 14.330 Project Site #2 B-1 (see page 3 of assignment handout) and Table 1. REQUIRED: Plot the effective stresses with depth. What is the effect of the downward water seepage on the effective stress compared to the static conditions? SOLUTION:

Downward flow causes a decrease in porewater pressure by izγγγγw, where i =

hydraulic gradient (0.05), z = depth below water table, γγγγw = unit weight of water.

Therefore, σσσσ'flow = σσσσ'static + izγγγγw (increase in pore pressure = decrease in effective stress). See Figure C for solution. PROBLEM #4 (30 Points): GIVEN: Figure 1 presents the general cross-section of two planned footings at your project site. Design loads of 108 kips for the columns and 6 kips per linear foot for the wall loads. Refer to Boring B-2 on Figure 1 for soil profile at both footing locations.

Figure 1. Planned Shallow Foundation Dimensions.

WALL FTG COLUMN FTG

4 ft 3 ft

EXISTING GND SURFACE

6 ft x 6 ft

3 ft




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Figure C. Total Stress, Pore Pressure, and Effective Stress with Depth (Downward

Flow). REQUIRED: From the provided information, determine the following:

• The change in vertical effective stresses under the center of the footings using Boussinesq, Westergaard, and 2V:1H methods in 0.5B increments to a depth of 5 times the footing width.

• The change in vertical stress distributions under the horizontal footing centerlines at depths of B and 2B to a distance of 4.5B away from the footing centerline.

• Provide a brief commentary on the differences between the methods for both footings.




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SOLUTION: SOLUTION: Determine applied footing stress q: Column Footing qcolumn = P/A = 108 kips/(36ft2) = 3 ksf. qcolumn = 3 ksf. Strip (i.e. Wall) Footing qstrip = P/A = [(12 kips/ft)(1 ft Unit Length)/[(4ft)(1ft Unit Length)] qstrip = 3 ksf. Change in Vertical Total Stresses: Change in total vertical stress with depth from Boussinesq and Westergaard methods determined from Pressure with Depth Charts provided in class and attached to the end

of this solution. 2V:1H Approximation based on the following formula: ∆σ = Q/[(B+z)(L+z)] (Equation A), where: Q = Foundation Load, B = Foundation Width, L = Foundation Length (unit length of 1 used for strip), and Z = depth below footing. See Tables A and B for calculations for the column and strip footings, respectively and Figure D for graphical representations. Table A. Summary Calculations for Column Footing.

Depth (B) Depth (ft) ∆σ∆σ∆σ∆σ Bouss.

(q)1 ∆σ∆σ∆σ∆σ Bouss.

(psf) ∆σ∆σ∆σ∆σ West.

(q)2 ∆σ∆σ∆σ∆σ West.

(psf) ∆σ∆σ∆σ∆σ 2V:1H

(psf)3

0.0 0.0 1 3000 1 3000 3000

0.5 3.0 0.68 2040 0.5 1500 1335

1.0 6.0 0.37 1110 0.22 660 750

1.5 9.0 0.2 600 0.13 390 480

2.0 12.0 0.11 330 0.074 220 335

2.5 15.0 0.077 230 0.049 145 245

3.0 18.0 0.047 140 0.035 105 190

3.5 21.0 0.04 120 0.026 80 150

4.0 24.0 0.033 100 0.019 55 120

4.5 27.0 0.025 75 0.017 50 100

5.0 30.0 0.019 55 0.014 40 85

NOTES: 1. From Boussinesq Pressure Distribution with Depth Chart. 2. From Westergaard Pressure Distribution with Depth Chart. 3. From Equation A.




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Table B. Summary Calculations for Strip Footing.

Depth (B) Depth (ft) Bouss.

(q)1 Bouss.

(psf) West. (q)2 West. (psf)

2V:1H (psf)3

0.0 0.0 1 2000 1 2000 2000

0.5 1.5 0.78 1560 0.59 1180 535

1.0 3.0 0.55 1100 0.4 800 250

1.5 4.5 0.4 800 0.28 560 145

2.0 6.0 0.31 620 0.22 440 95

2.5 7.5 0.26 520 0.17 340 65

3.0 9.0 0.22 440 0.15 300 50

3.5 10.5 0.18 360 0.135 270 40

4.0 12.0 0.16 320 0.12 240 30

4.5 13.5 0.144 290 0.099 200 25

5.0 15.0 0.132 265 0.09 180 20

NOTES: 1. From Boussinesq Pressure Distribution with Depth Chart. 2. From Westergaard Pressure Distribution with Depth Chart. 3. From Equation A.

Stresses at Planes B and 2B under Footings

Change in Total Vertical Stress (∆σ) can be determined using the Boussinesq or Westergaard charts provided in the lecture notes. Using the Boussinesq charts for the square footing, changes with stress away from the footing were calculated and are shown in Table C and Table D for the column and strip footings, respectively. Note that after distance of ~3B at a depth of 1B and a distance of ~4B at a depth of 2B, the change in vertical effective stress is negligible. Figures E and F provides these data in graphical form.




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Figure D. ∆σ with Depth (Boussinesq, Westergaard, and 2V:1H Methods).




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Table C. Summary Calculations at Planes below Footing of 1B and 2B for Column Footing (using Boussinesq Charts).

B x (ft) ∆σ∆σ∆σ∆σ/q @ z=B ∆σ∆σ∆σ∆σ @ z=B

(psf) ∆σ∆σ∆σ∆σ/q @ z=2B

∆σ∆σ∆σ∆σ @ z=2B (psf)

-4.5 -27 <0.001q 0 <0.001q 0

-4 -24 <0.001q 0 <0.001q 0

-3.5 -21 <0.001q 0 0.003 10

-3 -18 0.001 5 0.007 20

-2.5 -15 0.004 10 0.010 30

-2 -12 0.008 25 0.020 60

-1.5 -9 0.020 60 0.040 120

-1 -6 0.080 240 0.065 195

-0.5 -3 0.220 660 0.090 270

0 0 0.370 1110 0.110 330

0.5 3 0.220 660 0.090 270

1 6 0.080 240 0.065 195

1.5 9 0.020 60 0.040 120

2 12 0.008 25 0.020 60

2.5 15 0.004 10 0.010 30

3 18 0.001 5 0.007 20

3.5 21 <0.001q 0 0.003 10

4 24 <0.001q 0 <0.001q 0

4.5 27 <0.001q 0 <0.001q 0




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Table D. Summary Calculations at Planes below Footing of 1B and 2B for Strip Footing (using Boussinesq Charts).

B x (ft) ∆σ∆σ∆σ∆σ/q @ z=B ∆σ∆σ∆σ∆σ @ z=B

(psf) ∆σ∆σ∆σ∆σ/q @ z=2B

∆σ∆σ∆σ∆σ @ z=2B (psf)

-4.5 -13.5 <0.01q 0 <0.01q 0

-4 -12 <0.01q 0 0.012 25

-3.5 -10.5 <0.01q 0 0.018 35

-3 -9 <0.01q 20 0.030 60

-2.5 -7.5 0.014 30 0.050 100

-2 -6 0.025 50 0.081 160

-1.5 -4.5 0.070 140 0.140 280

-1 -3 0.200 400 0.210 420

-0.5 -1.5 0.450 900 0.280 560

0 0 0.550 1100 0.330 660

0.5 1.5 0.450 900 0.280 560

1 3 0.200 400 0.210 420

1.5 4.5 0.070 140 0.140 280

2 6 0.025 50 0.081 160

2.5 7.5 0.014 30 0.050 100

3 9 0.011 20 0.030 60

3.5 10.5 <0.01q 0 0.018 35

4 12 <0.01q 0 0.012 25

4.5 13.5 <0.01q 0 <0.01q 0

Brief Commentary: As shown in Figure D, the three methods compare reasonably well with depth for the column footing. However, for the strip footing, the 2V:1H method drastically under predicts the change in stress and therefore SHOULD NOT BE USED FOR STRIP FOOTINGS!




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Figure E. ∆σ under Footing - Column Footing.

Figure F. ∆σ under Footing - Strip Footing.




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PROBLEM #5 (10 Points): GIVEN: At the same project site, the project plans call for adding 5 ft of SP-SM fill over the entire site. This fill has a saturated unit weight of 120 pcf and a moist unit weight of 115 pcf after compaction to 98% of D1557. A surface parking lot will eventually be placed on top of the fill. Calculate and plot the total stresses, pore pressure, and effective stresses to the end of Boring B-2. Use the information provided in Problem #1 for your calculations. REQUIRED: Calculate and plot the total stresses, pore pressure, and effective stresses to the end of Boring B-2. Use the information provided in Problem #1 for your calculations. SOLUTION: Assume the five (5) feet of fill acts as a new soil layer. This is a valid assumption, since

no dimensions of the site were given. Therefore, the increase in total stresses (∆σ) is

equal to the moist unit weight of the SP-SM fill multiplied by the fill height (i.e. ∆σ = (γSP-

SM)(fill height) = (115 pcf)(5ft) = 575 psf). Since there is no change in pore water

pressure (∆u = 0) due to the added fill, the change in total stresses equals the change in

effective stresses (i.e. ∆σ = ∆σ'). Therefore, add ∆σ = (5ft)(115 pcf) = 575 psf to total and effective stresses calculated with depth in Problem #1. The solution is plotted in Figure F.




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Figure F. Total Stress, Pore Pressure, and Effective Stress with Depth (Static

Conditions with 6 ft of additional SP-SM Fill).

Documents

14.330 SOIL MECHANICS Assignment #5: Stresses in a Soil Mass.faculty.uml.edu/.../14.3302015Assignment5Solution.pdf · Assignment #5: Stresses in a Soil Mass. PROBLEM #1 (40 Points):