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chapter 2

Solving

2.2 A machine foundation can be idealized to a mass-spring system, as shown in Figure

Given

Weight of machine + foundation = 400 kN

Spring constant = 100,000 kN/m

Determine the natural frequency of undamped free vibration of this foundation and the

natural period.

2.3 Refer to Problem 2.2, What would be the static deflection zs of this foundation?

2.4 Refer to Example 2.3. For this foundation let time t = 0, z = z 0 = 0. z 0 = u0 = 0.

a. Determine the natural period T of the foundation.

b. Plot the dynamic force on the subgrade of the foundation due to the forced part of the

response for time t =0 to t = 2 T.

c. Plot the dynamic force on the subgrade of the foundation due to the free part of the

response for t = 0 to 2 T.

d. Plot the total dynamic force on the subgrade [that is, the algebraic sum of (b) and (c)].

Hint: Refer to Eq. (2.33)

: a حل

: b حل

t = 2Tتب t = 0

T = 0.1012 s , 2T = 0.2024 s

2.6 A foundation of mass m is supported by two springs attached in parallel (Figure

P2.6). Determine the natural frequency of the undamped free vibration.

2.7 For the system shown in Figure P2.7, calculate the natural frequency and period

given k1 =100 N/mm, k2 = 200 N/mm, k3 = 150 N/mm, k4 = 100 N/mm , k5 = 150

N/mm, and m = 100 kg.

K1 , k2 (سزی)

K4 , k5 (سزی)

K12 , k45 , k3 (هَاسی)

0

50

100

150

0 0.05 0.1 0.15 0.2 0.

dynamic force (KN)

Time , t(s)

total dynamic force on the subgrade

2.8 Refer to Problem 2.7. If a sinusoidally varying force Q = 50 sin w t (N) is applied to

the mass as shown, what would be the amplitude of vibration given w = 47 rad/s?

2.10 A machine foundation can be identified as a mass-spring system. This is subjected

to a forced vibration. The vibrating force is expressed as Q = Q0 sinw t

Q0 = 6.7 kN w = 3100 rad/min

Given

Weight of machine + foundation = 290 kN

Spring constant = 875 MN/m

Determine the maximum and minimum force transmitted to the subgrade.

Maximum force on the subgrade = 290 + 9.6 = 299.6 = 300 KN

Minimum force on the subgrade = 290 - 9.6 = 280.4 KN

2.14 A foundation weighs 800 kN. The foundation and the soil can be approximated as a

mass-spring-dashpot system as shown in Figure 2.2b.

Given

Spring constant = 200,000 kN/m

Dashpot coefficient = 2340 kN-s/m

Determine the following:

a. Critical damping coefficient cc.

b. Damping ratio

c. Logarithmic decrement

d. Damped natural frequency

: a حل

: b حل

: c حل

: d حل

Solving

chapter 3

-0.

-0.

0

1

0 2 4 6 8 10 12 14 16

U

(at depth z)/U (at z=0)

,

W(at depth z)/W (at z=0)

z

Variation of the amplitude of Rayleigh waves

Horizontal Component

Vertical Component

Solving

chapter 4

4.6 Repeat Problem 4.4 with the

following results. Also determine the

thickness of the second layer of soil

encountered.

[

]

y = 0.0042x - 5E- 05

y = 0.0004x + 0.

y = 0.0009x + 0.

0

0 20 40 60 80 100 120 140

Time (s)

Distance (m)

4.8 Refer to Figure 4.43 for the results of the following refraction survey: Determine: a. the P - wave

velocities in the two layers, b. z’ and z”, and c. the angle β.

y = 0.0032x + 0.

y = 0.0012x + 0.

0

0 20 40 60 80 100 120

Time (s)

Distance (m) , From Point A to Pont E

y = 0.0033x + 0.

y = 0.0003x + 0.

0

120 100 80 60 40 20 0

Time (s)

Distance (m) , From Point E to Point A

n x f L vr

4.12 A 20-m-thick sand layer in the field is underlain by rock.

The groundwater table is located at a depth of 5 m measured

from the ground surface. Determine the maximum shear

modulus of this sand at a depth of 10 m below the ground

surface. Given: void ratio = 0.6, specific gravity of soil solids =

2.68, angle of friction of sand = 36°. Assume the sand to be

round-grained.

0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.

L (m)

Velocity , vr , (m/s)

4.14 A remolded clay specimen was consolidated by a hydrostatic pressure of 205 kPa. The specimen

was then allowed to swell under a hydrostatic pressure of 105 kPa. The void ratio at the end of

swelling was 0.8. If this clay is subjected to a torsional vibration in a resonant column test, what

would be its maximum shear modulus ( G max)? These liquid and plastic limits of the clay are 58 and

28, respectively.

( )

4.16 Repeat Problem 4.15 given

H 1 = H 2 = H 3 = 6 m Gs (1) = Gs (2) = 2.

e 1 = 0.88    φ1 = 28°

e 2 = 0.68    φ 2 = 32°

PI of clay = 20

Estimate and plot the variation of the maximum shear

modulus ( G max) with depth for the soil profile.

Calculation of Effective Unit Weights

4.18 The unit weight of a sand deposit is 16.98 kN/m 3 at a relative density of 60%. Assume that, for

this sand φ= 30 + 0.15 RD where φis the drained friction angle and RD is the relative density (in

percent). At a depth of 6.09 m below the ground surface, estimate its shear modulus and damping

ratio at a shear strain level of 0.01%. Use the equation proposed by Seed and Idriss (1970).

4.20 For example 4.8, determine the damping ratio of the cemented sand.

Pa = 100 kPa, σ0 = 98 kPa and CC = 3%

DS = damping ratio of sand alone (%)

ΔDC = increase in the damping ratio due to cementation effect =

DCS = damping ratio of cemented sand (%)

Solving

chapter 5