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solution manual to soil dynamics

Typology: Study Guides, Projects, Research

2019/2020

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Download solution manual to soil dynamics and more Study Guides, Projects, Research Soil Mechanics and Foundations in PDF only on Docsity! io n chapter 2 Solving Page | 1 2.2 A machine foundation can be idealized to a mass-spring system, as shown in Figure 2.4. 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. : حل √ √ 2.3 Refer to Problem 2.2, What would be the static deflection zs of this foundation? : حل 2.4 Refer to Example 2.3. For this foundation let time t = 0, z = z0 = 0.z0 = 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 = 2T. c. Plot the dynamic force on the subgrade of the foundation due to the free part of the response for t = 0 to 2T. d. Plot the total dynamic force on the subgrade [that is, the algebraic sum of (b) and (c)]. Hint: Refer to Eq. (2.33) : aحل : bحل ( ( ) ) t = 0 تبt = 2T T = 0.1012 s , 2T = 0.2024 s Page | 4 2.8 Refer to Problem 2.7. If a sinusoidally varying force Q = 50 sin wt (N) is applied to the mass as shown, what would be the amplitude of vibration given w = 47 rad/s? : حل √ ( ) ( ) 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 sinwt 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. : حل √ √ Maximum force on the subgrade = 290 + 9.6 = 299.6 = 300 KN Minimum force on the subgrade = 290 - 9.6 = 280.4 KN Page | 5 2.12 A mass-spring system with two degrees of freedom is shown in Figure P2.12. Determine the natural frequencies wn1 and w n2 as a function of k1, k2, k3, m1, and m2. ̈ ̈ ̈ ̈ | | ( ) ( ) √ √ Page | 6 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 √ √ : bحل : cحل ( ) : dحل √ √ √ √ 91 آباى Page | 2 19آبان 2 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 12 14 16 U (a t d e p th z )/ U ( at z =0 ) , W (a t d e p th z )/ W ( at z =0 ) z Variation of the amplitude of Rayleigh waves Horizontal Component Vertical Component Solving chapter 4 Page | 1 4.2 A clayey soil specimen was tested in a resonant column device (torsional vibration; free–free end condition) for determination of shear modulus. Given: length of specimen = 90 mm, diameter of specimen = 35.6 mm, mass of specimen = 170 g, frequency at normal mode of vibration (n = 1) = 790 Hz. Determine the shear modulus of the specimen in kPa. ( ) 4.4 The results of a refraction survey in terms of time of first arrival (in milliseconds) and distance in meters is given below in tabular form. Assuming that the soil layers are perfectly horizontal, determine the Pwave velocities of the underlying soil layers and the thickness of the top layer. ( )( )( ) √ y = 0.0048x + 0.0124 y = 0.0018x + 0.0937 0 0.05 0.1 0.15 0.2 0.25 0.3 0 10 20 30 40 50 60 70 80 90 100 Ti m e ( s) Dsitance (m) is Page | 4 a. the P-wave velocities in the two layers [ ( ) ( )] ( ) ( ) b. z’and z” ( ) ( ) ( ) ( ) c. the angle β [ ( ) ( )] 4.10 The results of a subsoil exploration by steady-state vibration technique are given here (Section 4.15) Make necessary calculations and plot the variation of the wave velocity with depth. 𝑣𝑝 𝟑𝟎𝟖 𝒎 𝒔 Page | 5 n x f L vr 41 10 900 0.244 219.5 18 10 400 0.556 222.2 9 10 200 1.111 222.2 4.55 10 100 2.198 219.8 2.65 10 90 3.774 339.6 2.3 10 75 4.348 326.1 1.77 10 60 5.650 339.0 1.47 10 50 6.803 340.1 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. 0.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 L (m ) Velocity , vr , (m/s) Page | 6 ( ) ( ) ( )( ) ( ) ( ) ( ( )) ( ) √ 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 (Gmax)? These liquid and plastic limits of the clay are 58 and 28, respectively. ( ) ( ) √ 4.16 Repeat Problem 4.15 given H1 = H2 = H3 = 6 m Gs(1) = Gs(2) = 2.66 e1 = 0.88 φ1 = 28° e2 = 0.68 φ2 = 32° PI of clay = 20 Estimate and plot the variation of the maximum shear modulus (Gmax) with depth for the soil profile. ( ) Calculation of Effective Unit Weights ( ) ( ) ( ) Solving chapter 5 Page | 1 5.2 A concrete foundation is 2.5 m ×2 m in plan.. The foundation is supporting a machine. The total weight of the machine and the foundation is 270 kN. The machine imparts a vertical vibrating force Q = Q0 sinw t. Given Q0 = 27 kN (not frequency dependent). The operating frequency is 150 cpm. For the soil supporting the foundation, unit weight = 19.5 kN/m3, shear modulus = 45000 kPa. , and Poisson’ ratio = 0.3. Determine: a. resonant frequency, b. the amplitude of vertical vibration at resonant frequency, and c. the amplitude of vertical vibration at the operating frequency. ( ) a. resonant frequency √ [ √( ) ]√ ( √ ) b. the amplitude of vertical vibration at resonant frequency ( ( )) √ c. the amplitude of vertical vibration at the operating frequency √( ) √* ( ) + ( √ ) ( ) Page | 2 5.4 Consider the case of a single-cylinder reciprocating engine (Figure5.15a). For the engine, operating speed = 1000 cpm, crank (r1) = 90 mm,connecting rod (r2) = 350 mm, weight of the engine = 20 kN, andreciprocating weight = 65 N. The engine is supported by a concrete foundation block of 3 m × 2 m × 1.5 m (L B H). The unit weight of concrete is 23.58 kN/m3. The properties of the soil supporting the foundation are unit weight = 19 kN/m3, G = 24,000 kPa, and μ= 0.25. Calculate a. the resonant frequency, and b. the amplitude of vertical vibration at resonance. a. the resonant frequency √ ( ) √ ( )√ b. the amplitude of vertical vibration at resonance ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( √ ) Page | 5 5.10 Repeat Problem 5.9 assuming that the horizontal force is frequency dependent. The amplitude of the horizontal force at an operating frequency of 800 cpm is 40 kN. The weight of the machinery of the foundation is 100 kN. 5.9 Refer to Problem 5.7. Determine a. the resonant frequency for the sliding mode of vibration, and b. amplitude for the sliding mode of vibration at resonance. ssume the weight of the machinery on the foundation to be 100 kN. b. amplitude for the sliding mode of vibration at resonance G = 30,000 kPa, μ= 0.2, and = 1700 kg/m3. And operating frequency of 800 cpm Machinery weight = 100kN √ ( ) ( ) ( ) ( ) ( ) ( )( )( ) √ ( ) ( )√ a. the resonant frequency for the sliding mode of vibration √ ( ) ( )( ) ( ( )) √ is Page | 6 5.14 Solve Problem 5.13 with the following changes: Concrete foundation Length = 2 m Width = 1.5 m Height = 1.5 m , Depth of embedment, Df = 1.2 m Unit weight of concrete = 24 kN/m3 Vibrating machine Weight = 90 kN , frequency-dependent amplitude of vibrating force = 9 kN at an operating speed of 500 cpm Soil Gs = 22 MPa; G = 150 kPa; μ= 0.25 unit weight, ϒs = 18.5 kN/m3 (for side layer); unit weight, ϒ= 19.5 kN/m3 (below the base) Determine: a. damped natural frequency, b. amplitude of vertical vibration at resonance, and c. amplitude of vibration at operating speed. a. damped natural frequency √ ( ) ( ) √ ( ) ( √ ) ( √ ) √ √ √ b. amplitude of vertical vibration at resonance ( ) √ c. amplitude of vibration at operating speed √ ( ) ( ) is Page | 7 5.16 A horizontal piston-type compressor is shown in Figure P5.16. The operating is 800 cpm. The amplitude of the horizontal unbalanced force of the compressor is 25 kN. It creates a rocking motion of the foundation about O. The mass moment of inertia of the compressor assembly about the b’Ob’axis is 20×105 kg.m2. Determine: a. the natural frequency, and b. the amplitude of rocking vibration at resonance. Use the theory developed in Section 5.14. a. the natural frequency ( ) √ * ( )+ √ b. the amplitude of rocking vibration at resonance √ * ( )+ √ √ √ ( ) ( ) ( ) ( ) √ Page | 2 6.3 Redo Problem 6.1 with the following: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6.4 A rectangular foundation has a length L of 2.5 m. It is supported by a medium dense sand with a unit weight of 17 kN/m3. The sand has an angle of friction of 36º. The foundation may be subjected to a dynamic load of 735 kN increasing at a moderated rate. Using a factor of safety equal to 2, determine the width of the foundation. Use Df = 0.8 m. ( ) ( ) ( ) ( ) assume ( ) ( ) ( ) ( ) ( [ ]) ( ) ( [ ]) ( ) ( ) ( ) Page | 3 Check : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Try 1 : assume B=0.8 m ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6.5 A foundation 2.25 m square is supported by saturated clay. The unit weight of this clay is 18.6 kN/m3. The depth of the foundation is 1.2 m. Determine the ultimate bearing capacity of this foundation assuming that the load will be applied very rapidly. Given the following for the clay [laboratory unconsolidated-undrained triaxial (static) test results]: Undrained cohesion, cu = 90 kPa Strain-rate factor = 1.4 ̇ Page | 4 ( ) ( ) [ ( )] ( ) 6.6 Redo Problem 6.5 with the following changes: Foundation width = 1.5 m Foundation length = 2.6 m Foundation depth = 1.75 m ̇ ( ) [ ( )] [ ( )] ( ) 6.7 A clay deposit has an undrained cohesion (static test) of 90 kPa. A static field plate load test was conducted with a plate having a diameter of 0.5 m. When the load per unit area q was 200 kPa, the settlement was 20 mm. a. Assume that, for a given value of q, settlement is proportional to the width of the foundation. Estimate the settlement of a prototype circular foundation in the same clay with a diameter of 3 m (static loading). b. The strain-rate factor of the clay is 1.4. If a vertical transient load pulse were applied to the foundation as given in part (a), what would be the maximum transient load (in kN) that will produce the same maximum settlement (Smax) as calculated in part (a)? a Page | 2 8.4 Refer to Problem 8.3. Where would be the location of the resultant if the wall is rotating about its top? ̅ ( ) 8.6 Redo Problem 8.1 using the modified Culmann graphical solution procedure. is Page | 3 8.8 For the retaining wall and the backfill given in Problem 8.1, determine the passive force PPE per unit length of the wall. 8.10 Consider a 3.6 m high vertical retaining wall (β= 0°) with a horizontal backfill (i = 0°). Given for the soil are φ= 32°, = 19.5 kN/m3, and δ= 0. a. Calculate PAE and the location of resultant with ku = 0.1and kh = 0.15. b. For the results of (a), what should be the weight of the wall per meter length for no lateral movement? The factor of safety against sliding is 1.4. c. What should be the weight of the wall for an allowable lateral displacement of 25 mm? Given Au = Aa 0.15; the factor of safety against sliding is 1.4. a. Calculate PAE and the location of resultant with ku = 0.1and kh = 0.15 ( ) b.Weight of wall without lateral movment [ ( ) ] ( ) c.Weight of wall with allowable lateral displacement of 25mm ( ) ( )( ) [ ( ) ] ( )
Solving
aT ame Ay
Page | 3 10.6 Consider the soil and the groundwater table conditions given in Problem 10.2. Assume that the relative density in the field is 60%. The maximum expected intensity of ground shaking (amax/g) is 0.2 and the magnitude of earthquake is 7.5. a. Calculate and plot the variation of the shear stress t au induced in the sand deposit with depth 0 – 21 m. Use Eq. (10.26). b. Calculate the variation of the shear stress required to cause liquefaction with depth. Plot the shear stress determined in the same graph as used in (a). Use Eq. (10.23). c. From the plotted graph, determine the depth at which liquefaction is initiated. Mean grain size (D50) = 0.2 mm ,Depth of water table = 3 m ,Unit weight of soil above G.W.T = 17 kN/m3 ,Unit weight of soil below G.W.T = 19.5 kN/m3 a. Z ϒ ϒh CD τ , a(max)/g=0.2 0 17 0.0 1.00 0.00 1 17 17.0 1.00 2.21 2 17 34.0 0.99 4.38 3 17 51.0 0.98 6.50 4 19.5 51.0 0.97 6.43 5 19.5 70.5 0.96 8.80 6 19.5 109.5 0.95 13.52 7 19.5 168.0 0.93 20.31 8 19.5 246.0 0.90 28.78 9 19.5 343.5 0.87 38.85 10 19.5 460.5 0.85 50.89 11 19.5 597.0 0.83 64.42 12 19.5 753.0 0.81 79.29 13 19.5 928.5 0.77 92.94 14 19.5 1123.5 0.73 106.62 15 19.5 1338.0 0.70 121.76 16 19.5 1572.0 0.68 138.96 17 19.5 1825.5 0.66 156.63 18 19.5 2098.5 0.65 177.32 19 19.5 2391.0 0.63 195.82 20 19.5 2703.0 0.60 210.83 21 19.5 3034.5 0.57 224.86 0 5 10 15 20 25 0.00 100.00 200.00 300.00 D e p th ( m ) Part a Part a Page | 4 b. Z ϒ ϒh Cr=0.61 τ , RD2=60% 0 17 0.0 1.00 0.00 1 17 17.0 1.00 2.68 2 17 34.0 0.99 5.35 3 17 51.0 0.98 8.03 4 9.69 51.0 0.97 8.03 5 9.69 60.7 0.96 9.55 6 9.69 80.1 0.95 12.60 7 9.69 109.1 0.93 17.18 8 9.69 147.9 0.90 23.28 9 9.69 196.4 0.87 30.90 10 9.69 254.5 0.85 40.05 11 9.69 322.3 0.83 50.73 12 9.69 399.8 0.81 62.93 13 9.69 487.1 0.77 76.65 14 9.69 584.0 0.73 91.90 15 9.69 690.5 0.70 108.68 16 9.69 806.8 0.68 126.98 17 9.69 932.8 0.66 146.80 18 9.69 1068.5 0.65 168.15 19 9.69 1213.8 0.63 191.03 20 9.69 1368.8 0.60 215.43 21 9.69 1533.6 0.57 241.35 0 5 10 15 20 25 0.00 100.00 200.00 300.00 D e p th ( m ) Part b Part b Page | 5 c. 0 5 10 15 20 25 0.00 50.00 100.00 150.00 200.00 250.00 300.00 D e p th ( m ) Part a Part b Zone of initial liquefaction