Fundamentals of Geomechanics - Transtutors · School of Civil & Environmental Engineering Fundamentals of Geomechanics 4 ASSIGNMENT 2 - STRESSES IN DRY SOIL 1- A deep deposit of dry

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CVEN9525

Fundamentals of Geomechanics

Assignments

The University of New South Wales CVEN9525 School of Civil & Environmental Engineering Fundamentals of Geomechanics

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ASSIGNMENT 1 - PHASE RELATIONSHIPS AND SOIL CLASSIFICATION 1- The aim of compacting a fill is to fit as many soil solids into a given volume as possible.

Two soils are available as potential fill, one is classified as SW and the other SP. Which would make a better fill? Briefly explain why.

2- A clay sample is compacted at a water content of = 14%. After the compaction, the

degree of saturation is 73.5%. Calculate the dry unit weight of the soil.

3- A soil sample has a void ratio of 0.6 and a degree of saturation of 75%. Determine: a) The total unit weight of the soil sample. b) The shrinkage limit of the soil.

4- A trench 80 m long by 0.45 m wide by 1m deep is to be excavated in a clay soil at a site. An

undisturbed cylindrical sample of the clay, with a diameter of 35 mm and length of 70 mm, is taken prior to digging the trench. The original mass of the sample is 101.5 g where mass of the sample after oven drying is 67.3 g. When the trench is dug the excavated clay will be stockpiles and then used as trench backfill. The clay is to be compacted into the trench in uniform depth layers at a total density of 1.8 t/m3 and a moisture content of 14%. After all the excavated clay is compacted into the trench what depth of the trench will still remain to be filled?

5- A rectangular excavation measuring 20 m by 40 m in plan is taken down to a depth of 3 m.

The sides of the excavation have vertical faces. A cylindrical sample of the soil, with a diameter of 50 mm and height of 100 mm, is taken prior to excavation. The soil sample has an original mass of 363 g and mass of sample after oven drying is 310 g. The excavated soil is to be used to backfill a trench 1 m wide by 2 m deep. The soil is to be compacted into the trench at a moisture content of 24% such that its dry density is 1.6 t/m3. a) What would be the shrinkage limit of the soil in the tube sample? b) What length of trench could be backfilled by the excavated soil? c) What would be the degree of saturation of the compacted soil in the trench?

6- Classify soils A, B and C according to the USCS.

Soil A Soil B Soil C % Finer than 4.75mm sieve 60 80 45 % Finer than 0.075mm sieve 4 60 10 Coefficient of uniformity 5.2 - 4.8 Coefficient of curvature 1.4 - 2.2 Liquid limit NP 65 40 Plastic limit NP 45 20

Answers: Q1: SW Q2: 17.26 kN/m3 Q3: a) 19kN/m3, b) 22.6% Q4: 0.367m Q5: a) 25.6%, b) 1185m, c) 97% Q6: SP, MH, GW-GC


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ASSIGNMENT 2 - STRESSES IN DRY SOIL 1- A deep deposit of dry gravel has a void ratio of 0.62. Determine the vertical stress at a point

4m below the surface of the gravel. 2- A level soil deposit has the following soil profile: 0 - 3m Dry gravel d = 20kN/m3 Ko = 0.3 3 - 6m Dry sand d = 17kN/m3 Ko = 1.4

Determine the major principal stress at a depth of 5m in this deposit. 3- A section through an inclined tunnel in soil is shown in

Figure 1. At Point X, on the roof of the tunnel, the stress acting in the vertical direction is equal to 300 kPa. The normal and shear stresses acting on the plane of the tunnel at Point X, as shown in the figure, are both zero. a) Draw the Mohr circle for the state of stress at Point X. b) What is the value of the major principal stress at Point X? c) What angle does the plane of major principal stress make with the horizontal?

4- An element of soil is subjected to the normal compressive stresses of 100 and 40kPa as shown in Figure 2. Shear stresses are now applied to the planes on which these stresses act. What is the magnitude of the shear stress which must be applied to those planes such that the plane of major principal stress in the element is horizontal?

5- The stresses shown in Figure 3 are applied at a point

in a soil mass. Determine the angle that the plane of major principal stress makes with the x-axis

Answers: Q1: 64kPa Q2: 131.6kPa Q3: b) 400kPa, c) 30o Q4: 51.96kPa Q5: 67.5o

Point X 0

60o

0 Soil

Tunnel shown by dottel line

100 kPa

30o

Vertical

Horizontal

40 kPa

Figure 3

Figure 2

Figure 1

30 80

45 o

z

x


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ASSIGNMENT 3 - STRESSES IN SOIL 1- A level site consists of a deep deposit of gravel having a dry density of 1.7t/m3. The water

table is 2 m below the surface of the gravel. During a period of wet weather the site is flooded such that 2 m of water covers the site. Determine the change in vertical effective stress at a point 4m below the surface of the gravel as a result of this flooding. The height of capillary rise in gravel is zero.

2- The soil profile at a site consists of: 0 – 8m Sand d = 16 kN/m3 sat = 19 kN/m3 Ko = 0.5 8 – 12m Clay d = 18 kN/m3 sat = 20 kN/m3 Ko = 0.3 The water table is initially at a depth of 1 m below the surface. The water table is to be rapidly lowered to a depth of 6 m and maintained at that level permanently. The height of capillary rise in the sand is assumed to be zero. Consider two points in the soil profile, Point A at a depth of 4 m and Point B at a depth of 10 m. Determine the effective vertical and horizontal stresses at Points A and B for two cases: a) Initial conditions where the water table is 1 m below the surface. b) A long time after lowering the water table to 6 m below the surface. 3- A level site has the following soil profile with the water table 4 m below the ground: 0 - 3m Dry gravel t = 16kN/m3 3 - 5m saturated sand t = 17kN/m3 5 - 8m saturated clay t = 20kN/m3 8m + Rock The following construction sequence is then carried out. The uppermost 2m of gravel is excavated and a very large diameter oil storage tank is built at the new surface. The depth of oil in the tank is 6m and the specific gravity of the oil is 0.8. The mass of steel in the tank is negligible compared to the mass of oil in the tank. Determine the change in vertical effective stress caused by construction at the bottom of the clay layer. 4- The total stresses shown in Figure 1 are applied at a point in a soil mass. The pore pressure in the soil at the point is u. Determine what value u would need to take if the minor principal effective stress at the point was to be zero. 5- Calculate the change in the vertical stress at a point 1.5m below point x due to application of 100 kPa pressure on the rectangular footing shown in the figure opposite. Do by: a) 2:1 method b) Boussinesq method (Fadum’s Chart) c) Newmark’s chart (based on Boussinesq method) Answers: Q1: -12.6 kPa Q2: a) = 43.6 , = 21.8 , = 100.8 , = 30.2 b) = 64.0 ,

= 32.0 , = 134.8 , = 40.4 Q3: 15 kPa Q4: 40.9 kPa Q5: a) 0 b) 3 kPa c) 3 kPa

30 50

45o

z

x

u

Figure 1

1m 10m

2m

1m x


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ASSIGNMENT 4 - ONE-DIMENSIONAL SEEPAGE 1- Water is seeping through an inclined aquifer

as shown in Figure 1. Two standpipes are installed at the mid-depth of the aquifer at Points x and y. The water rises 3.9m up the tube at Point x and 4.1 m at Point y. The permeability of the aquifer soil is l0-5m/s. What is the rate of flow in the aquifer in m3/s per metre run of the aquifer?

2- Water is flowing through a two layers filter

system as shown in Figure 2. The first stage of the filter system is a silty sand having a coefficient of permeability of 2×10-4m/s. The area of the filter normal to the direction of flow is 0.2m2. When the inlet pressure is 49kPa the flow through the system is measured to be 2×10-5m3/s. Determine the coefficient of permeability of the second stage filter soil.

3- A long lake is located in a range of hills. Beneath the lake there is a weathered dyke inclined

at 60o to the horizontal which connects to a gravel aquifer, a section through which is shown in Figure 3. The surrounding rock mass may be regarded as impermeable. Tests on gravel aquifer indicate that it has a coefficient of permeability of 0.15m/s and flow measurement at the aquifer outlet spring gave a value of 0.015 m3/s per metre run (into the page).

a) Determine the coefficient of permeability of the weathered dyke. b) An inclined borehole is to be installed as shown in the figure. Determine the RL of the water level in the borehole a long time after drilling the hole.

Answers: Q1: 1.67×10-6m3/s Q2: 1.43×10-5m/s Q3: a) 9.62×10-3m/s, b) 63.6m

Figure 3

`

Weathered dyke

RL: 90 m

RL: 55 m

2m

60o

120m

RL: 40 m Spring outlet

1.5m

Inclined bore holeRL: 100 m

Gravel aquifer

1m

Stage 2 soil

1m

49kPa

x 0.5m

Silty sand

Figure 2

1m 10o Aquifer

1m

3.9m x

y30m

Impermeable soil

4.1m

Figure 1


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ASSIGNMENT 5 – TWO-DIMENSIONAL SEEPAGE 1- A long coffer dam with a drainage trench in its centre is to be constructed in a shallow lake, a

section through which is shown in the figure below. The cofferdam is kept dry by pumping water from the drainage trench. a) Draw the flow net for the steady state seepage flow through the soil around the dam. You may take advantage over the symmetry of the problem and draw the flow net for only half of the cofferdam. b) From the flow net determine the pore water pressure at Point A located 2 m above the impermeable bedrock. c) If the soil has a coefficient of permeability of 10-3 m/s, what capacity of the pump (in m3/s) would be required if a pump is to be placed every 20 m along the trench.

A

Impervious wall

5m

Impermeable rock

3m

2m

2m

2m

1m

.5m

Water

Soil

Not to scale

2- A long drain pipe is installed below ground as shown in the figure below. The flow net for

the steady stage seepage from the ground surface to the pipe during a rain fall is shown in the figure. The soil is a silty sand having a coefficient of permeability of 5×10-5m/s and a total unit weight of 18 kN/m3. a) What is the vertical effective stress in the soil at Point X? b) At what rate, in m3/s per metre run of the pipe, does the pipe collect the rainwater seepage?

Answers: Q1: b) ~ 38kPa, c) ~0.1m3/s Q2: a) 38.1kPa, b) 1.8×10-4 m3/s

Point X

Rainfall

3m2.5m


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ASSIGNMENT 6 - ONE-DIMENSIONAL SETTLEMENT 1- As part of a construction project, a 7.5 m thick layer of clay is to be loaded with a temporary

3 m thick sand layer, as shown in the figure. The figure shows the water-table location, soil unit weights, and the compression curve properties for the clay. Assume the sand layer remains dry. (a) Calculate the value of in the middle of the clay layer (at 3.75 m below the water table) before the sand layer is applied, and after consolidation is complete. (b) Based on your answer in part (a), and the compression curve characteristics, calculate the settlement that will occur under these conditions. (c) How much will the clay layer heave when the 3 m sand layer is removed? (Holtz et al.)

2- A sample of clay is subjected to a consolidation test. The sample is initially seated under a

pressure of 20 kPa. At the end of the seating stage the height of the sample is measured as 20.4 mm. The pressure is then increased to 40 kPa which causes a final settlement of 2.3 mm in the sample. At the completion of this increment the moisture content of the sample is 22%. a) If it is assumed that the sample underwent virgin compression during the pressure increment, what is the value of the compression index, Cc, of the clay? b) At the completion of the 20 to 40 kPa load increment the sample is unloaded back to a pressure of 20 kPa resulting in a final heave of 0.8 mm. What is the value of the rebound index, Cr, of the clay? c) At the completion of the unloading stage in part b, the pressure is increased to 80 kPa. What would be the value of void ratio in the specimen at the end of the 80 kPa increment?

3- A level site consists of the following soil profile: 0 – 6 m Gravel 6 – 8 m Clay 8+ Bedrock

The water table is at a depth of 2 m. Tests have shown that if the gravel is above water table it will be dry with a d = 17 kN/m3 while if it is below it will be saturated with t = 22 kN/m3. Testing of a sample of clay taken from the mid-depth of the layer gave the following properties: t = 20 kN/m3 w = 31% pc = 150 kPa

Cc = 0.6 Cr = 0.2 a) Development at the site involves permanently lowering the water table by 2 m and placing fill at a unit weight of 22 kN/m3 on top the profile. What depth of fill will need to be placed over the site in order to produce a total final settlement of the clay layer of 65 mm? b) A long time after lowering the water table and filling to the depth found in part a, a loading of 100 kPa is placed over the surface of the fill. Determine the final thickness of the clay layer a long time after loading.

Answers: Q1: a) Before applying the sand layer: 40.1 ; After completing the consolidation: 88.1 b) 16.2 c) 7.7 Q2: a) 0.668, b) 0.232 c) 0.3819 Q3: a) 2.63m, b) 1.796m


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ASSIGNMENT 7 - CONSOLIDATION AND RATE OF CONSOLIDATION 1- The soil profile at a site is as follows: 0 – 2 m Sand t = 20 kN/m3 2 – 7 m Clay t = 18 kN/m3 7 m + Impermeable rock

The water table is 1 m below the surface of the sand. A sample of the clay taken from the mid-depth of the clay layer has the following properties:

w = 24% pc = 110 kPa Cc = 0.4 Cr = 0.1 cv = 1.98×10-7m2/s a) A 3 m layer of gravel fill is now placed over the surface of the site (fill = 21 kN/m3) and left in place for a long time. Determine the thickness of the clay layer 2 years after placing the fill. b) A long time after the fill was in place and all the consolidation settlements were completed the uppermost 2 m of the fill is removed from the site. Determine the thickness of the clay layer a long time after removing the 2 m of fill. You may assume that the conditions at the sampling point are representative of the entire layer. A one-point calculation may be made based on conditions at the representative point.

2- A site consists of a 4 m deep layer of clay underline by impermeable rock. The clay has a

total unit weight of 20 kN/m3. A piesometer is installed in the clay such that it measures the pore pressure in the clay at a depth of 1 m below the surface. The site is suddenly subjected to a surface surcharge. The following piesometer reading are obtained:

Pressure before surcharging: 9 kPa Pressure immediately after surcharging: 91 kPa Pressure 100 days after surcharging: 45 kPa

a) Determine the coefficient of consolidation, cv, of the clay. b) Find the vertical effective stress at a point 3 m below the surface 200 days after surcharging. c) How long (in days) would it take for 90% settlement of the clay layer to occur due to surcharging.

3- A 4m deep clay layer is underlain by impermeable shale. A sample of clay from the layer is

subjected to an appropriate increment of stress in a consolidation test giving the following results:

Sample height at the start of increment: 55mm. Sample height 2 minutes after the start of increment: 53mm. Sample height at the end of increment: 48mm.

If gravel fill is placed over the clay layer how long would it take for 80% consolidation of the clay to occur?

Answers: Q1: a) 4.908m, b) 4.941m Q2: a) 1.85×10-7m2/s, b) 54.4kPa, c) 800days Q3: ~240 days,


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ASSIGNMENT 8 – SHEAR STRENGTH (1) 1- A sample of loose sand having a u1t of 30o is subjected to a minor principal total stress of

100kPa and a pore pressure of 20kPa. What value of major principal total stress would need to be applied to the sample to cause failure?

2- A level soil deposit consists of a deep layer of sand with the following properties: t = 20kN/m3 Ko = 0.5 = 30o

The water table is 1m below the surface. Consider a point x which is 5m below the surface. What value of shear stress would need to be applied on a plane passing through x and making an angle of 30o with the horizontal to cause shear failure of the sand on that plane?

3- In a direct shear test on a specimen of cohesionless sand, the vertical normal stress on the specimen is 240 kN/m2 and the horizontal shear stress at failure is 160 kN/m2. (a) Assuming uniform stress distribution within the failure zone and a straight line failure envelope which goes through the origin, determine by means of the Mohr circle the magnitude and direction of the principal stresses at failure. (b) Find the angle between the shear plane and the major principle plane. (c) Explain why it is not possible to determine the principal stresses in a direct shear specimen for an applied horizontal shear stress which is not large enough to cause failure. (After A. Casagrande)

4- A series of direct shear tests using a small shear box (40 mm high) to be conducted on sand

to investigate the way in which water affects its shear strength. In each of the tests described below the normal pressure used during the test was held constant at 100 kPa. a) A sample of dry sand is placed in the shear box and the shear stress at failure is measured as 70 kPa. What is the angle of internal friction of the sand? b) A sample of the sand is just saturated by pouring water into the box until it flows away at the surface. What would be the value of the shear stress at failure in this specimen? c) A sample of the sand is placed in the shear box and pore pressure in the sample is controlled by allowing water to seep steadily upward through the sand. The system is set so that the pore pressure in the sample is held constant at 50 kPa. What would be the value of the shear stress at failure in this specimen? d) The entire shear box is immersed in water at a depth of 5 m. What would be the value of the shear stress at failure in this specimen?

Answers: Q1: 260 kPa Q2: 17.5kPa Q3: a) = 539.0 angled at 28.2° with the horizontal; =

154.4 angled at 61.8° with the horizontal b) 61.8° Q4: a) 35o, b) 70kPa, c) 35kPa, d) 70kPa


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ASSIGNMENT 9 - SHEAR STRENGTH (2) 1- A footing is constructed on a soil as shown in the

opposite figure. When the footing exerts a pressure q on the surface of the soil, the stresses acting on an element of the underlying soil are as shown in the figure. The underlaying soil has a cohesion of 20 kPa and a friction angle of 30o. Sketch the Mohr circle for the stress state in the element when it is at the state of failure. You may ignore any stresses due to the self weight of the soil. At what level of q the element of soil will fail?

2- A soil profile at a site consists of a 5 m deep layer of clay over permeable rock. The surface

of the clay is horizontal and the water table is at a depth of 1 m. The clay has the following properties: t = 20 kN/m3 Ko = 0.5 cv = 1.4×10-7 m2/s Tests indicate that the undrained cohesion of the clay at a point in the layer is a function of the effective stress state as defined by cu = (v + h) / 2 a) Consider Point X which is located at a depth of 3 m in the clay layer. Determine the in-situ value of cu at Point X. b) The site is to be subjected to filling with 2.5 m of fill placed over the soil surface at a t = 18.4 kN/m3. Consolidation due to the fill will increase the undrained cohesion. How long will it take for the cohesion at Point X to reach a value of 50 kPa?

3- A UU triaxial test is carried out on a sample of saturated normally consolidated clay. During

the test the pore pressure in the specimen is measured. At failure the following values are noted: Deviatoric stress: 180 kPa Cell pressure: 100 kPa Pore pressure: 30 kPa a) What value of cu is indicated by this test for the clay? b) If it is assumed that c is zero for this clay, what value of is indicated by this test? c) A sample of this clay is to be loaded very slowly in a direct shear test using a normal pressure of 100 kPa. What would be the shear stress at failure in the clay? d) The clay is to be subjected to a UU test using a cell pressure of 200 kPa. What value of pore pressure would you expect to measure in the clay at failure?

4- A consolidated drained triaxial test on a sample of clay gives the following results: Cell pressure (kPa): 50 100 150 Deviator stress at failure (kPa): 105 178 251

a) Find c' and ' for this clay b) An identical sample of this clay is subjected to an unconsolidated undrained triaxial test using a cell pressure of 100kPa. It is found that the deviator stress at failure is 40kPa. What would be the average value of pore pressure in the specimen as it fails?

Answers: Q1: 194.6kPa Q2: a) 30.3kPa b) 211days Q3: a) 90kPa b) 34.2o c) 68kPa d) 130kPa Q4: a) 10kPa and 25o b) 94.5kPa

Pressure q

0.9q

0.2q

0.1q


12

ASSIGNMENT 10 - SLOPE STABILITY 1- Surface of a granular soil mass is inclined at 25o to the horizontal. The soil is saturated

throughout with a moisture content of 15.8 percent, particle specific gravity of 2.65 and a friction angle of 38o. At the depth of 1.83m water is seeping through the soil parallel to the surface. Determine the safety factor against sliding on a plane parallel to the surface at a depth of 3.05m below the surface ( = 10 ⁄ ). What would be the safety factor for the same plane if the level of the seeping water lifted up to the surface of the soil?

3- A long vertical drain is going to be made in a clay soil without any support. Assume the failure plane shown in the opposite figure and evaluate the safety factor of the cut assuming the following properties for the soil:

= 35 , = 0 , = 18

3- A 12m deep slope is cut into homogeneous soft clay with a saturated unit weight of 16 kN/m3 and undrained shear strength of 24 kPa. The slope angle is 30o to the horizontal and the geometry of the potential slop surface passing through the head and toe of the slope is given in the opposite figure. For the slip surface shown, calculate the undrained factor of safety. The centroid of the sliding mass is located at a distance 0.8259R from the centre of the circle, where R is the radius (Exam question, 2011).

Answers: Q1: 1.36, 0.9 Q2: 1.3 Q3: 1.18

6 m

45o

13

Assignments Check List - C Correct solution Your solution Check the box if correct Assignment 1:

Q1: SW Q2: 17.26 kN/m3 Q3: a) 19kN/m3, b) 22.6% Q4: 0.367m Q5: a) 25.6%, b) 1185m, c) 97% Q6: SP, MH, GW-GC

Assignment 2:

Q1: 64 kPa Q2: 131.6 kPa Q3: b) 400kPa, c) 30o Q4: 51.96kPa Q5: 67.5o

Assignment 3:

Q1: -12.6 kPa Q2: a) = 43.6, = 21.8, = 100.8, = 30.2 b) = 64.0, = 32.0, = 134.8, = 40.4(in kPa) Q3: 15kPa Q4: 40.9kPa Q5: 3kPa

Assignment 4: Q1: 1.67×10-6m3/s Q2: 1.43×10-5m/s Q3: a) 9.62×10-3m/s, b) 63.6m

Assignment 5: Q1: b) ~ 38kPa, c) ~0.1m3/s Q2: a) 38.1kPa, b) 1.8×10-4m3/s

Assignment 6: Q1: a) 40.1 ; 88.1 , b) 16.2 , c) 7.7 Q2: a) 0.668, b) 0.232, c) 0.3819 Q3: a) 2.63m, b) 1.796m

Assignment 7: Q1: a) 4.908m, b) 4.941m Q3: a) 1.85×10-7 m2/s, b) 54.4 kPa, c) 800 days Q3: ~ 240 days

Assignment 8: Q1: 260 kPa Q2: 17.5 kPa Q3: a) 539.0 , 28.2°; 154.4 61.8°; b) 61.8° Q4: a) 35o, b) 70kPa, c) 35kPa, d) 70kPa

Assignment 9: Q1: 194.6 kPa Q2: a) 30.3 kPa, b) 211 days

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Q3: a) 90 kPa, b) 34.2o, c) 68 kPa, d) 130 kPa Q4: a) 10kPa and 25o, b) 94.5kPa

Assignment 10: Q1: 1.36, 0.9 Q2: 1.3 Q3: 1.18

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Fundamentals of Geomechanics - Transtutors · School of Civil & Environmental Engineering Fundamentals of Geomechanics 4 ASSIGNMENT 2 - STRESSES IN DRY SOIL 1- A deep deposit of dry