14.330 soil mechanics assignment #5: stresses in a soil...

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

One University Avenue Lecturer

Lowell, Massachusetts 01854 PA105D

Tel: (978) 934-2621 Fax: (978) 934-3052

e-mail: [email protected]

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

DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING

14.330 2014 Assignment 5 Solution of 11

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

PROBLEM #1 (20 Points): GIVEN: You are a staff engineer for a local geotechnical engineering firm. As part of the geotechnical exploration for a project, several subsurface tests have been conducted. You are given the results of one traditional soil boring with Standard Penetration Testing (SPT). The results of this boring, labeled as B-2, are presented on page 3 of this assignment. Based on testing of collected soil samples, the encountered soils have the unit weights listed in Table 1. Table 1. Summary of Soil Unit Weights from Boring B-2.

UCSC Symbol γγγγ (pcf) γγγγsat (pcf)

SP (Upper) 103 108

ML 103 109

CH 110 115

SM 114 117

SP (Lower) 114 120

CL 121 125

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 the 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.

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

One University Avenue Lecturer

Lowell, Massachusetts 01854 PA105D

Tel: (978) 934-2621 Fax: (978) 934-3052

e-mail: [email protected]

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

DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING


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

Conditions). PROBLEM #2 (10 Points): GIVEN: You have just learned that a hydraulic gradient of 0.10 upwards exists at your project site. REQUIRED: Using this information, recalculate the effective stresses from the ground surface to the bottom of the borehole (i.e. a depth of 75 ft). Plot these effective stresses with depth. Briefly explain the effect of the upward water seepage on the effective stress compared to the static conditions.


SOLUTION:

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

hydraulic gradient (0.1), 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). PROBLEM #3 (10 Points): GIVEN: You have just learned that a hydraulic gradient of 0.10 downwards exists at your project site.


REQUIRED: Using this information, recalculate the effective stresses from the ground surface to the bottom of the borehole (i.e. a depth of 75 ft). Plot these effective stresses with depth. Briefly explain 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 (0.10), 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.

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

Flow).


PROBLEM #4 (50 Points): GIVEN: Figure 1 presents the general cross-section of two planned footings at your project site.

Figure 1. Planned Shallow Foundation Dimensions (NTS).

The project structural engineer has given you design loads of 108 kips for the columns and 12 kips per linear foot for the wall loads. Refer to Boring B-2 on page 3 of the assignment handout for the soil profile at both footing locations. 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.

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.

WALL FTG COLUMN FTG

3 ft 5 ft

EXISTING GND SURFACE

6 ft x 6 ft

4 ft


See Tables B and C for calculations for the column and strip footings, respectively and Figure D for graphical representations. Table B. 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 1565

1.0 6.0 0.37 1110 0.22 660 825

1.5 9.0 0.2 600 0.13 390 510

2.0 12.0 0.11 330 0.074 220 345

2.5 15.0 0.077 230 0.049 145 250

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 80

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

Table C. 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 3000 1 3000 3000

0.5 2.0 0.78 2340 0.59 1770 400

1.0 4.0 0.55 1650 0.4 1200 170

1.5 6.0 0.4 1200 0.28 840 95

2.0 8.0 0.31 930 0.22 660 60

2.5 10.0 0.26 780 0.17 510 40

3.0 12.0 0.22 660 0.15 450 30

3.5 14.0 0.18 540 0.135 405 25

4.0 16.0 0.16 480 0.12 360 20

4.5 18.0 0.144 430 0.099 295 15

5.0 20.0 0.132 395 0.09 270 10

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


Figure D. ∆σ with Depth.

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 D and Table E 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.


Table D. 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 -27.00 -27 <0.001q 0 <0.001q

-4 -24.00 -24 <0.001q 0 0.001

-3.5 -21.00 -21 <0.001q 0 0.003

-3 -18.00 -18 <0.001q 0 0.006

-2.5 -15.00 -15 0.004 10 0.010

-2 -12.00 -12 0.008 25 0.020

-1.5 -9.00 -9 0.021 65 0.040

-1 -6.00 -6 0.081 245 0.067

-0.5 -3.00 -3 0.230 690 0.090

0 0.00 0 0.360 1080 0.110

0.5 3.00 3 0.230 690 0.090

1 6.00 6 0.081 245 0.067

1.5 9.00 9 0.021 65 0.040

2 12.00 12 0.008 25 0.020

2.5 15.00 15 0.004 10 0.010

3 18.00 18 <0.001q 0 0.006

3.5 21.00 21 <0.001q 0 0.003

4 24.00 24 <0.001q 0 0.001

4.5 27.00 27 <0.001q 0 <0.001q


Table E. 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 -18.00 -18 <0.01q 0 <0.01q

-4 -16.00 -16 <0.01q 0 0.013

-3.5 -14.00 -14 <0.01q 0 0.018

-3 -12.00 -12 <0.01q 0 0.030

-2.5 -10.00 -10 <0.01q 0 0.045

-2 -8.00 -8 0.015 45 0.080

-1.5 -6.00 -6 0.035 105 0.130

-1 -4.00 -4 0.070 210 0.205

-0.5 -2.00 -2 0.200 600 0.280

0 0.00 0 0.550 1650 0.320

0.5 2.00 2 0.200 600 0.280

1 4.00 4 0.070 210 0.205

1.5 6.00 6 0.035 105 0.130

2 8.00 8 0.015 45 0.080

2.5 10.00 10 <0.01q 0 0.045

3 12.00 12 <0.01q 0 0.030

3.5 14.00 14 <0.01q 0 0.018

4 16.00 16 <0.01q 0 0.013

4.5 18.00 18 <0.01q 0 <0.01q

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!


Figure E. ∆σ under Footing - Column Footing.

Figure F. ∆σ under Footing - Strip Footing.


PROBLEM #5 (10 Points): GIVEN: At the same project site, the project plans call for adding 6 ft of SP-SM fill over the entire site. This fill has a saturated unit weight of 120pcf 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. 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 (6) 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)(6ft) = 690 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 = (6ft)(115 pcf) = 690 psf to total and effective stresses calculated with depth in Problem #2. The solution is plotted in Figure F.

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

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

14.330 soil mechanics assignment #5: stresses in a soil...

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