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Question 1
ప్రశ్న 1
Which combination of assumptions is fundamental to the classical Boussinesq solution for stress increments caused by a vertical point load applied at the surface of a soil mass?
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Explanation:
Boussinesq theory represents the ground as a homogeneous and isotropic elastic half-space. The material is assumed to obey linear elastic stress-strain relationships. The soil mass extends infinitely in the horizontal directions and downward from a horizontal surface. A concentrated vertical load acts at the surface, producing additional stresses within the medium. Actual layered or strongly anisotropic ground may depart from these ideal assumptions and require a different analysis.
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Question 2
ప్రశ్న 2
A foundation applies an additional load to soil that already carries stresses from the weight of overlying deposits. Which interpretation of the Boussinesq vertical-stress solution is correct?
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Explanation:
The Boussinesq solution evaluates stresses introduced by an externally applied load. It does not directly calculate the original geostatic stress arising from the weight of soil layers. The computed vertical-stress increment can be added to the initial vertical total stress when linear superposition is appropriate. Effective stress also depends on pore-water pressure and the drainage response of the soil. Consequently, stress distribution, effective-stress change and bearing capacity are related but distinct engineering calculations.
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Question 3
ప్రశ్న 3
A vertical concentrated surface load Q = 200 kN acts on an ideal Boussinesq soil half-space. What is the vertical stress increment at a depth z = 2 m directly below the load? Use delta_sigma_z = 3Q/(2 pi z²) on the load axis and pi = 3.1416.
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Explanation:
The radial distance is zero because the observation point lies directly below the point load. The Boussinesq equation therefore reduces to delta_sigma_z = 3Q/(2 pi z²). Substitution gives 3(200)/[2(3.1416)(2²)]. The calculated vertical-stress increment is approximately 23.87 kPa. Hence the correct rounded value is 23.9 kPa.
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Question 4
ప్రశ్న 4
A vertical surface point load of 400 kN acts on an ideal homogeneous elastic half-space. Find the vertical stress increment at a point 2 m below the surface and 2 m horizontally from the load axis. Use delta_sigma_z = [3Q/(2 pi z²)]/[1 + (r/z)²]^(5/2), with pi = 3.1416.
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Explanation:
The observation point has depth z = 2 m and radial distance r = 2 m. Therefore r/z = 1, and the radial correction denominator is 2^(5/2). The on-axis stress for the same load and depth is 3(400)/[2 pi(2²)] = approximately 47.75 kPa. Dividing by 2^(5/2), or approximately 5.6569, gives 8.44 kPa. The off-axis stress is smaller than the corresponding on-axis stress because of lateral stress dispersion.
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Question 5
ప్రశ్న 5
For a fixed vertical point load on the surface of an ideal Boussinesq half-space, the vertical stress increment is measured at depth z directly below the load. What happens to this increment when the observation depth is doubled to 2z?
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Explanation:
Directly below a point load, Boussinesq vertical stress is proportional to Q/z². The load Q remains unchanged in this comparison. Doubling the depth changes the denominator from z² to 4z². The calculated stress increment therefore becomes one-quarter of the original value. This inverse-square relationship applies specifically along the vertical load axis of the stated point-load model.
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Question 6
ప్రశ్న 6
At the same depth z beneath a vertical surface point load, compare the Boussinesq vertical-stress increment at radial distance r = z with the increment directly on the load axis. What is the ratio delta_sigma_z(r = z)/delta_sigma_z(r = 0)? Take 2^(5/2) = 5.6569.
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Explanation:
The Boussinesq radial variation factor is [1 + (r/z)²]^(-5/2). At r = 0, this factor equals unity. At r = z, it becomes 2^(-5/2). Dividing one by 5.6569 gives approximately 0.1768. Thus the vertical-stress increment at radial distance equal to the depth is about 17.68% of the on-axis value.
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Question 7
ప్రశ్న 7
Two equal surface point loads of 200 kN each are located symmetrically on opposite sides of a subsurface observation point. The observation point is 2 m below ground level and 2 m horizontally from the axis of each load. Assuming Boussinesq conditions and linear superposition, what is the combined vertical stress increment? Use pi = 3.1416.
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Explanation:
Each load is 200 kN, and the observation point has z = 2 m and r = 2 m relative to each load. The Boussinesq equation gives approximately 4.22 kPa from one load at that point. Linear elastic superposition permits the stress increments from independently applied loads to be added. The combined increment is therefore 4.22 + 4.22 = approximately 8.44 kPa. Symmetry makes both contributions equal but does not eliminate their vertical-stress effects.
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Question 8
ప్రశ్న 8
Which statement correctly compares the role of elastic constants in the classical Boussinesq point-load solution?
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Explanation:
The classical Boussinesq expression for vertical stress contains the applied point load and the observation-point coordinates. It does not explicitly contain Young's modulus or Poisson's ratio. The vertical-stress increment follows from equilibrium and the elastic half-space assumptions for the prescribed surface load. Calculating displacement or settlement requires additional material stiffness information. Therefore a correct stress-distribution result alone does not determine the foundation settlement.
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Question 9
ప్రశ్న 9
A flexible rectangular footing applies a uniform net pressure q = 180 kPa. The Boussinesq influence factor for the specified observation point directly beneath one corner, obtained for the footing dimensions and observation depth, is I = 0.24. What is the vertical stress increment at that point?
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Explanation:
The vertical-stress increment for a uniformly loaded area is commonly written as delta_sigma_z = qI. Here q is the uniform net applied pressure, and I is the dimensionless influence factor for the stated geometry and observation point. Multiplying gives 180(0.24) = 43.2 kPa. The coefficient accounts for the dimensions and position of the loaded area relative to the observation point. No additional chart is required because the appropriate influence factor is supplied.
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Question 10
ప్రశ్న 10
For a flexible rectangular footing carrying uniform pressure q = 150 kPa, the vertical stress is required directly beneath its centre. Each of the four equal rectangular quadrants contributes an influence factor of 0.20 when the centre is treated as its corner. What is the combined vertical stress increment?
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Explanation:
The full rectangular loaded area can be divided into four equal quadrants around its centre. The observation point lies beneath the common corner of those four quadrants. Each quadrant has the supplied influence factor of 0.20. Linear superposition gives a total influence factor of 4(0.20) = 0.80. The combined vertical-stress increment is therefore 150(0.80) = 120 kPa.
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Question 11
ప్రశ్న 11
An L-shaped uniformly loaded area is represented as a large rectangle minus an unloaded rectangular cutout. At the same observation point and depth, the influence factors for the large rectangle and the cutout are 0.32 and 0.08, respectively. If the applied pressure is 200 kPa over the L-shaped area, what is the vertical stress increment?
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Explanation:
Influence factors for loaded areas can be combined using elastic superposition. The L-shaped area is obtained by subtracting the unloaded cutout from the larger rectangle. Its net influence factor is therefore 0.32 - 0.08 = 0.24. Multiplication by the uniform pressure gives delta_sigma_z = 200(0.24) = 48 kPa. The subtraction is valid because both supplied influence factors refer to the same observation point and loading reference.
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Question 12
ప్రశ్న 12
A uniformly loaded flexible circular area has radius R = 2 m and surface pressure q = 100 kPa. What is the vertical stress increment at depth z = 2 m directly beneath its centre? Use delta_sigma_z = q[1 - {1 + (R/z)²}^(-3/2)] and 2^(3/2) = 2.8284.
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Explanation:
The radius-to-depth ratio is R/z = 2/2 = 1. The circular-area expression becomes delta_sigma_z = 100[1 - 2^(-3/2)]. The supplied value of 2^(3/2) is 2.8284, whose reciprocal is approximately 0.3536. Therefore the stress increment is 100(1 - 0.3536) = 64.64 kPa. The appropriate rounded answer is 64.6 kPa.
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Question 13
ప్రశ్న 13
A flexible circular footing of radius 2 m applies a uniform net contact pressure of 160 kPa. Calculate the vertical stress increment at depth 4 m directly beneath its centre using the Boussinesq circular-area expression.
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Explanation:
The radius-to-depth ratio is R/z = 2/4 = 0.50. The centreline circular-area expression is delta_sigma_z = q[1 - {1 + (R/z)²}^(-3/2)]. Substitution gives 160[1 - (1.25)^(-3/2)]. The stress influence factor is approximately 0.28446. Multiplication by 160 kPa gives approximately 45.5 kPa.
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Question 14
ప్రశ్న 14
A uniformly loaded circular area of radius R = 2 m produces a vertical stress increment directly beneath its centre. At what depth z does this increment decrease to 50% of the applied uniform pressure? Use 2^(2/3) - 1 = 0.5874.
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Explanation:
The circular-area influence factor is 1 - [1 + (R/z)²]^(-3/2). For a stress increment equal to half the applied pressure, the influence factor must equal 0.50. Rearranging gives (R/z)² = 2^(2/3) - 1 = 0.5874. Thus z = R/sqrt(0.5874). With R = 2 m, the calculated depth is approximately 2.61 m.
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Question 15
ప్రశ్న 15
An infinitely long flexible strip footing is uniformly loaded at q = 100 kPa over its full width B = 4 m. Calculate the vertical stress increment at depth z = 2 m beneath the centreline using delta_sigma_z = (2q/pi)[arctan(b/z) + bz/(b² + z²)], where b = B/2, arctan(1) = pi/4 and pi = 3.1416.
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Explanation:
The strip half-width is b = 4/2 = 2 m. Because z = 2 m, the ratio b/z equals one and the arctangent is pi/4. The second term is bz/(b² + z²) = 4/8 = 0.50. Therefore the bracketed expression is pi/4 + 0.50. Multiplying by 2(100)/pi gives approximately 81.8 kPa.
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Question 16
ప్రశ్న 16
A rectangular footing of width B = 2 m and length L = 4 m carries uniform net pressure q = 150 kPa. Using the approximate 2:1 stress-spread method, what is the vertical stress increment at depth z = 2 m below the foundation base? Use delta_sigma_z = qBL/[(B + z)(L + z)].
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Explanation:
The approximate 2:1 method assumes that the load spreads over an enlarged rectangular area with depth. The effective dimensions at depth z are B + z and L + z. For the stated footing, these dimensions are 4 m and 6 m. The estimated stress is 150(2)(4)/[(4)(6)] = 50 kPa. This is an approximate stress-spread result rather than an exact integrated Boussinesq solution.
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Question 17
ప్రశ్న 17
A 2 m by 3 m footing transfers a total net vertical load of 600 kN uniformly to the foundation level. Under the approximate 2:1 load-spread method, what is the average vertical stress increment at depth z = 3 m beneath its base?
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Explanation:
The net total load is 600 kN and remains constant in the stated approximate distribution model. At 3 m depth, the assumed load-spread dimensions are 2 + 3 = 5 m and 3 + 3 = 6 m. Their product is 30 m². Dividing the load by this enlarged area gives 600/30 = 20 kPa. This calculation estimates the average stress over the enlarged distribution area and does not reproduce every local stress predicted by elasticity theory.
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Question 18
ప్రశ్న 18
A geotechnical engineer constructs a stress-bulb representation beneath a uniformly loaded foundation. Which interpretation of the plotted isobars is most appropriate?
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Explanation:
Stress bulbs are graphical representations of stress distribution beneath loaded areas. Their isobars connect points at which the calculated stress increment has the same specified value. These contours help indicate the depth and lateral extent of foundation load influence. They do not directly show soil strength, stratigraphic boundaries or groundwater levels. When evaluating settlement, the engineer considers which compressible layers lie within regions experiencing significant stress increases.
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Question 19
ప్రశ్న 19
Two nearby flexible foundations produce overlapping vertical-stress influence zones. At a particular subsurface point, Foundation A applies a uniform net pressure of 140 kPa with influence factor 0.25, while Foundation B applies 120 kPa with influence factor 0.15. Assuming linear elastic superposition, what is the combined vertical stress increment?
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Explanation:
Under linear elastic conditions, stress increments from separately applied loads can be superimposed. Foundation A contributes 140(0.25) = 35 kPa at the observation point. Foundation B contributes 120(0.15) = 18 kPa at the same point. Their combined contribution is 35 + 18 = 53 kPa. The result demonstrates why adjacent foundations may influence stresses and settlement beyond their individual footprints.
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Question 20
ప్రశ్న 20
A foundation is placed at a depth Df = 2 m below the original ground surface. The excavated soil had unit weight gamma = 18 kN/m³, and the completed foundation applies a gross contact pressure of 220 kPa at its base. Assuming the removed overburden is represented by gamma Df, what is the stress increment at a selected point below the base if the given influence factor for the net pressure is 0.40?
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Explanation:
The initial overburden pressure removed by excavation is gamma Df. Its value is 18(2) = 36 kPa. The net foundation pressure relative to the original ground stress is therefore 220 - 36 = 184 kPa. Multiplying this net pressure by the supplied influence factor gives 184(0.40) = 73.6 kPa. Using gross pressure without accounting for the removed overburden would overstate the net stress increment under the stated assumptions.
వివరణ:
Question 21
ప్రశ్న 21
At a point in saturated soil, the initial vertical effective stress is 90 kPa. A foundation subsequently produces an additional vertical total stress of 48 kPa, and sufficient drainage occurs so that the long-term pore-water pressure at this point is unchanged. What is the final vertical effective stress?
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Explanation:
Effective stress is the difference between total stress and pore-water pressure. The foundation introduces an additional vertical total stress of 48 kPa. Under the specified drained condition, pore-water pressure does not change between the initial and final states. Therefore the entire 48 kPa increment is transferred to the effective stress. The final effective stress is 90 + 48 = 138 kPa.
వివరణ:
Question 22
ప్రశ్న 22
An initially saturated clay deposit has vertical effective stress of 80 kPa at a selected point. Immediately after foundation loading, the vertical total stress has increased by 60 kPa while pore-water pressure has increased by 48 kPa. What are the increase in vertical effective stress and its new value at that point?
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Explanation:
The effective-stress increment equals the total-stress increment minus the pore-pressure increment. The applied foundation load increases vertical total stress by 60 kPa. At the specified instant, pore-water pressure rises by 48 kPa. Thus the vertical effective-stress increment is 60 - 48 = 12 kPa. Adding this to the initial effective stress gives 80 + 12 = 92 kPa.
వివరణ:
Question 23
ప్రశ్న 23
A uniform net foundation pressure of 200 kPa acts over a flexible loaded area. The average vertical-stress influence factor across a compressible soil layer is estimated as 0.20, and the layer has thickness H = 3 m and coefficient of volume compressibility mv = 4.0 × 10^-4 kPa^-1. Assuming one-dimensional compression and constant mv, what is the estimated eventual settlement of this layer?
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Explanation:
The average vertical-stress increment in the compressible layer is qI = 200(0.20) = 40 kPa. Under the stated one-dimensional compression assumption, settlement is S = mv DeltaSigma H. Substitution gives S = (4.0 × 10^-4)(40)(3) m. This equals 0.048 m. Therefore the estimated eventual layer settlement is 48 mm.
వివరణ:
Question 24
ప్రశ్న 24
A rigid foundation bears on a highly stratified soil deposit consisting of stiff sand layers, soft clay and shallow bedrock. Which statement most appropriately describes the limitations of directly applying a uniform-pressure Boussinesq half-space analysis?
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Explanation:
Boussinesq theory provides an idealized elastic solution for a homogeneous and isotropic soil mass. Strong contrasts in stiffness between adjacent strata can alter the actual stress and deformation response. A rigid foundation may also develop a nonuniform contact-pressure distribution depending on the foundation and ground characteristics. Shallow bedrock violates the assumption of a semi-infinite soil mass. Therefore a more representative layered elastic, numerical or otherwise appropriate geotechnical analysis may be needed for reliable design.
వివరణ:
Question 25
ప్రశ్న 25
A raft foundation applies a uniform net pressure of 200 kPa. Two underlying compressible layers are evaluated using average vertical-stress influence factors: Layer 1 has H1 = 2 m, mv1 = 3.0 × 10^-4 kPa^-1 and I1 = 0.35; Layer 2 has H2 = 3 m, mv2 = 4.0 × 10^-4 kPa^-1 and I2 = 0.15. Assuming one-dimensional primary compression with constant mv values, what is the combined eventual settlement?
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Explanation:
The average stress increment in Layer 1 is 200(0.35) = 70 kPa. Its settlement is (3.0 × 10^-4)(70)(2) = 0.042 m, or 42 mm. The average stress increment in Layer 2 is 200(0.15) = 30 kPa. Its settlement is (4.0 × 10^-4)(30)(3) = 0.036 m, or 36 mm. Adding the two layer settlements gives 42 + 36 = 78 mm, demonstrating how stress distribution and soil compressibility jointly determine settlement.
వివరణ:
Answer Key సమాధానాల పట్టిక
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Question 1 ప్రశ్న 1Answer: B. The soil is homogeneous, isotropic, linearly elastic and represented by a semi-infinite continuum. సమాధానం: B.
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Question 2 ప్రశ్న 2Answer: D. It gives the additional vertical total-stress component caused by the applied loading, which must be combined with existing stress conditions as appropriate. సమాధానం: D.
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Question 3 ప్రశ్న 3Answer: A. 23.9 kPa. సమాధానం: A.
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Question 4 ప్రశ్న 4Answer: C. 8.44 kPa. సమాధానం: C.
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Question 5 ప్రశ్న 5Answer: B. It becomes one-quarter of its original value. సమాధానం: B.
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Question 6 ప్రశ్న 6Answer: A. 0.1768. సమాధానం: A.
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Question 7 ప్రశ్న 7Answer: D. 8.44 kPa. సమాధానం: D.
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Question 8 ప్రశ్న 8Answer: C. The vertical stress increment for the stated point-load half-space is independent of E and Poisson's ratio, although deformation calculations generally require elastic properties. సమాధానం: C.
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Question 9 ప్రశ్న 9Answer: B. 43.2 kPa. సమాధానం: B.
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Question 10 ప్రశ్న 10Answer: A. 120 kPa. సమాధానం: A.
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Question 11 ప్రశ్న 11Answer: D. 48 kPa. సమాధానం: D.
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Question 12 ప్రశ్న 12Answer: C. 64.6 kPa. సమాధానం: C.
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Question 13 ప్రశ్న 13Answer: A. 45.5 kPa. సమాధానం: A.
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Question 14 ప్రశ్న 14Answer: B. 2.61 m. సమాధానం: B.
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Question 15 ప్రశ్న 15Answer: D. 81.8 kPa. సమాధానం: D.
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Question 16 ప్రశ్న 16Answer: C. 50 kPa. సమాధానం: C.
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Question 17 ప్రశ్న 17Answer: A. 20 kPa. సమాధానం: A.
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Question 18 ప్రశ్న 18Answer: B. Each isobar represents a contour of equal calculated vertical stress increment due to the applied load. సమాధానం: B.
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Question 19 ప్రశ్న 19Answer: D. 53 kPa. సమాధానం: D.
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Question 20 ప్రశ్న 20Answer: A. 73.6 kPa. సమాధానం: A.
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Question 21 ప్రశ్న 21Answer: C. 138 kPa. సమాధానం: C.
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Question 22 ప్రశ్న 22Answer: B. 12 kPa increase; final effective stress 92 kPa. సమాధానం: B.
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Question 23 ప్రశ్న 23Answer: D. 48 mm. సమాధానం: D.
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Question 24 ప్రశ్న 24Answer: C. The ideal homogeneous elastic half-space and uniform contact-pressure assumptions may be inadequate, so actual layering and foundation rigidity should be considered in selecting a suitable analysis. సమాధానం: C.
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Question 25 ప్రశ్న 25Answer: D. 78 mm. సమాధానం: D.