Please keep at least one language enabled. || కనీసం ఒక భాషను ఎంచుకోండి.
Question 1
ప్రశ్న 1
A prismatic member is subjected simultaneously to an axial load P through the centroid and a bending moment M about a centroidal principal axis. Within linear-elastic theory, which expression gives the normal stress at a fibre a signed distance y from the neutral centroidal axis?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The axial load through the centroid produces a uniform direct stress P/A over the section. The bending moment produces a linearly varying normal stress My/I. In the linear-elastic small-deformation range, these stresses may be superposed algebraically. One extreme fibre receives an increase in the direct stress while the opposite fibre receives a decrease. Therefore Option B is correct.
వివరణ:
Question 2
ప్రశ్న 2
A compressive force P acts parallel to the longitudinal axis of a member but at an eccentricity e from the centroidal axis. Which statically equivalent loading system acts at the centroid?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
An eccentric axial force may be transferred to the centroid if its moment about the centroid is retained. The force itself remains equal to P and acts through the centroid in the equivalent system. Moving its line of action through the distance e introduces a couple of magnitude Pe. The resulting section therefore experiences both direct stress and bending stress. Hence Option C is correct.
వివరణ:
Question 3
ప్రశ్న 3
A rectangular section of width b and depth d carries a compressive load P at an eccentricity e measured along the depth direction from the centroid. Taking compressive stress magnitudes as positive, which pair gives the extreme-fibre stresses?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The direct compressive stress is P/A, where A = bd. The eccentricity produces a bending moment Pe about the centroidal axis parallel to the width. The rectangular section modulus for this bending is Z = bd²/6. Therefore the extreme bending stress is Pe/Z = (P/A)(6e/d). Adding and subtracting this from the direct stress gives Option A.
వివరణ:
Question 4
ప్రశ్న 4
A compressive axial load passes exactly through the centroid of a homogeneous prismatic section and no bending moment is applied. Which stress distribution results, neglecting local load-introduction effects?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A centroidal axial load has zero eccentricity with respect to every centroidal axis. Consequently it produces no bending moment about the centroid. Under the elementary direct-stress assumption, every point of the cross-section carries the same normal stress. The stress magnitude is P divided by the cross-sectional area A. Therefore Option D is correct.
వివరణ:
Question 5
ప్రశ్న 5
A rectangular column section is 200 mm wide and 300 mm deep. A compressive load of 300 kN acts with an eccentricity of 40 mm from the centroid along the 300 mm depth direction. What are the maximum and minimum normal stresses?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The area is 200 × 300 = 60000 mm², so the direct compressive stress is 300000/60000 = 5 MPa. The section modulus about the relevant centroidal axis is bd²/6 = 3.0 × 10^6 mm³. The eccentric moment is 300000 × 40 = 12 × 10^6 N·mm, producing an extreme bending stress of 4 MPa. The two edge stresses are therefore 5 + 4 = 9 MPa and 5 - 4 = 1 MPa, both compressive. Hence Option B is correct.
వివరణ:
Question 6
ప్రశ్న 6
A rectangular section of depth 240 mm is under eccentric compression about its centroidal axis parallel to the width. The measured extreme stresses are 12 MPa compression and 4 MPa compression. What is the eccentricity of the load from the centroid?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The average of the two extreme stresses is the direct stress, giving (12 + 4)/2 = 8 MPa. Half their difference is the extreme bending stress, giving (12 - 4)/2 = 4 MPa. Thus the ratio of bending stress to direct stress is 4/8 = 0.5. For a rectangle this ratio is 6e/d, so 6e/240 = 0.5. Solving gives e = 20 mm, making Option C correct.
వివరణ:
Question 7
ప్రశ్న 7
A rectangular section of depth d carries a compressive load eccentric only along that depth direction. What is the limiting eccentricity from the centroid for which no tensile stress develops anywhere across the section?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The minimum stress is (P/A)(1 - 6e/d). To avoid tension, this minimum stress must be zero or compressive. Therefore 1 - 6e/d must be greater than or equal to zero. This requires e to be no greater than d/6. Hence Option A is correct.
వివరణ:
Question 8
ప్రశ్న 8
What does the middle-third rule mean for a rectangular section subjected to uniaxial eccentric compression about one centroidal axis?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For a rectangle, the kern limit in one bending direction is d/6 on either side of the centroid. The total allowable central interval is therefore d/3. A compressive resultant passing anywhere within this interval produces no tensile stress at either extreme edge under the linear stress model. At the kern boundary, the stress at one edge becomes exactly zero. Therefore Option B correctly describes the middle-third rule.
వివరణ:
Question 9
ప్రశ్న 9
A rectangular section has width b in the x direction and depth d in the y direction. A compressive load is applied with eccentricities ex and ey from the centroid. Which condition places the load within or on the kern so that no tensile stress occurs at any corner?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Biaxial eccentricity creates bending about both centroidal axes of the rectangle. At the least-compressed corner, the two bending reductions act together. Normalizing the x eccentricity by b/6 and the y eccentricity by d/6 gives the no-tension criterion. The resulting kern is a rhombus centered at the centroid rather than a rectangle. Therefore Option D is correct.
వివరణ:
Question 10
ప్రశ్న 10
A solid circular section of diameter D carries an eccentric compressive load. What is the maximum eccentricity from the centroid for which the entire section remains in compression?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The kern distance in a given direction is Z/A. For a solid circle, Z = πD³/32 and A = πD²/4. Their ratio is D/8. Therefore a compressive load must remain within a concentric circular kern of radius D/8 to avoid tensile stress. Hence Option C is correct.
వివరణ:
Question 11
ప్రశ్న 11
A solid circular column of diameter 200 mm carries a compressive load of 400 kN at an eccentricity of 20 mm. What are the maximum and minimum normal stresses approximately?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The direct compressive stress is 4P/(πD²), which is approximately 12.73 MPa. For a solid circular section, the ratio of extreme bending stress to direct stress is 8e/D. Here that ratio is 8 × 20/200 = 0.8. The extreme stresses are therefore 12.73(1 + 0.8) = 22.92 MPa and 12.73(1 - 0.8) = 2.55 MPa. Both are compressive, so Option A is correct.
వివరణ:
Question 12
ప్రశ్న 12
A rectangular section is 250 mm wide and 400 mm deep. A compressive load acts with eccentricity 40 mm along the 400 mm depth. If the maximum permissible compressive stress is 12 MPa, what is the maximum load based on combined direct and bending stress?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The section area is 250 × 400 = 100000 mm². For the stated eccentricity, the maximum stress is (P/A)(1 + 6e/d). The factor is 1 + 6 × 40/400 = 1.6. Setting 1.6P/100000 equal to 12 N/mm² gives P = 750000 N. Therefore the maximum load is 750 kN and Option D is correct.
వివరణ:
Question 13
ప్రశ్న 13
A rectangular section 100 mm wide and 200 mm deep carries an eccentric tensile load of 200 kN at an eccentricity of 50 mm along the depth. Taking tensile stress as positive, what are the extreme stresses?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The area is 20000 mm², so the uniform tensile stress is 10 MPa. The section modulus is bd²/6 = 666666.7 mm³. The bending moment is 200000 × 50 = 10 × 10^6 N·mm, giving an extreme bending stress of 15 MPa. Thus one edge has 10 + 15 = 25 MPa tension while the opposite edge has 10 - 15 = -5 MPa, which is 5 MPa compression. Therefore Option B is correct.
వివరణ:
Question 14
ప్రశ్న 14
A compressive load on a rectangular section acts exactly at the kern boundary for bending about one centroidal axis. What is the corresponding normal-stress distribution across the depth?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
At the kern boundary, the eccentricity equals d/6 for the relevant rectangular depth. The bending stress at the least-compressed edge then exactly equals the direct compressive stress in magnitude. Consequently the stress at that edge becomes zero. The opposite edge carries twice the average direct compressive stress, and the variation between the edges remains linear. Therefore the distribution is triangular and Option C is correct.
వివరణ:
Question 15
ప్రశ్న 15
For a rectangular section under eccentric compression at the middle-third limit, the stress distribution is triangular from zero at one edge to maximum compression at the other. At what distance from the maximum-compression edge does the resultant compressive force act?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The resultant of a triangular stress distribution passes through the centroid of the stress diagram. The centroid of a triangle lies one-third of its base dimension from the side having the maximum ordinate. Here the base dimension is the section depth d. Thus the resultant acts d/3 from the maximum-compression edge. This is equivalent to an eccentricity d/6 from the section centroid, so Option A is correct.
వివరణ:
Question 16
ప్రశ్న 16
A rectangular section of depth 300 mm is subjected to eccentric compression about the centroidal axis parallel to its width. The extreme stresses are 10 MPa compression and 2 MPa compression. What is the load eccentricity from the centroid?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The direct stress is the average of the two edge stresses, which is 6 MPa. The extreme bending stress is half their difference, which is 4 MPa. Therefore the ratio of bending stress to direct stress is 4/6 = 2/3. For a rectangular section this ratio equals 6e/d. Solving 6e/300 = 2/3 gives e = 33.3 mm, so Option D is correct.
వివరణ:
Question 17
ప్రశ్న 17
A solid circular section has diameter 240 mm and carries a compressive load of 300 kN at an eccentricity of 30 mm. What are the extreme stresses approximately?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For a 240 mm solid circle, the kern-limit eccentricity is D/8 = 30 mm. The load therefore acts exactly on the kern boundary. The direct compressive stress is 4P/(πD²) = approximately 6.63 MPa. At the kern boundary, the bending stress at an extreme fibre has the same magnitude as the direct stress. Thus the extreme stresses are about 13.26 MPa compression and zero, making Option B correct.
వివరణ:
Question 18
ప్రశ్న 18
A rectangular section is 300 mm wide in the x direction and 400 mm deep in the y direction. A 600 kN compressive load acts at ex = 20 mm and ey = 20 mm from the centroid. What are the maximum and minimum corner stresses?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The direct stress is 600000/(300 × 400) = 5 MPa compression. The normalized bending contribution from ex is 6ex/b = 0.4, corresponding to 2 MPa at an extreme x edge. The contribution from ey is 6ey/d = 0.3, corresponding to 1.5 MPa at an extreme y edge. At the most compressed corner the contributions add to give 5 + 2 + 1.5 = 8.5 MPa, while at the opposite corner they subtract to give 1.5 MPa compression. Therefore Option C is correct.
వివరణ:
Question 19
ప్రశ్న 19
A 300 mm wide by 400 mm deep rectangular section carries a 600 kN compressive load at ex = 30 mm and ey = 40 mm from the centroid. What are the maximum and minimum corner stresses?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The direct stress is 5 MPa compression. The x-eccentricity contributes 5(6 × 30/300) = 3 MPa at the extreme x coordinates. The y-eccentricity similarly contributes 5(6 × 40/400) = 3 MPa at the extreme y coordinates. At the corner nearest the load line, the stress is 5 + 3 + 3 = 11 MPa compression. At the diagonally opposite corner the result is 5 - 3 - 3 = -1 MPa, meaning 1 MPa tension, so Option A is correct.
వివరణ:
Question 20
ప్రశ్న 20
For a section under uniaxial eccentric compression, let c be the distance from the centroidal bending axis to the critical extreme fibre. Which expression gives the kern distance k from the centroid in that direction?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The minimum stress at the extreme fibre is P/A - Pec/I. At the kern boundary this stress is exactly zero. Cancelling P and solving for e gives e = I/(Ac). Since section modulus Z is I/c, the same expression is Z/A. Therefore Option D is correct.
వివరణ:
Question 21
ప్రశ్న 21
A rectangular section 150 mm wide and 300 mm deep carries a centroidal compressive load of 450 kN together with an independent bending moment of 18 kN·m about the centroidal axis parallel to the width. What are the extreme normal stresses?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The section area is 150 × 300 = 45000 mm², giving a direct compressive stress of 10 MPa. The section modulus is 150 × 300²/6 = 2.25 × 10^6 mm³. The bending stress from 18 kN·m is 18 × 10^6/2.25 × 10^6 = 8 MPa. The edge stresses are therefore 10 + 8 = 18 MPa and 10 - 8 = 2 MPa, both compressive. Hence Option B is correct.
వివరణ:
Question 22
ప్రశ్న 22
The same 150 mm by 300 mm rectangular section carries a centroidal compressive load of 450 kN, but the applied bending moment is increased to 27 kN·m. What are the extreme stresses?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The direct compressive stress remains 10 MPa. The section modulus remains 2.25 × 10^6 mm³. A bending moment of 27 kN·m produces an extreme bending stress of 12 MPa. One edge therefore carries 10 + 12 = 22 MPa compression, while the other has 10 - 12 = -2 MPa. The negative result represents 2 MPa tension, so Option C is correct.
వివరణ:
Question 23
ప్రశ్న 23
A solid circular section of diameter 200 mm carries an eccentric compressive load. The minimum compressive stress is required to be at least one-quarter of the maximum compressive stress. What is the maximum permissible eccentricity?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For a solid circle, let r = 8e/D denote the ratio of extreme bending stress to direct stress. The extreme compressive stresses are proportional to 1 + r and 1 - r. The condition (1 - r)/(1 + r) = 1/4 gives 4 - 4r = 1 + r. Thus r = 0.6, and e = 0.6D/8 = 0.075D = 15 mm for D = 200 mm. Therefore Option A is correct.
వివరణ:
Question 24
ప్రశ్న 24
Consider the following statements about direct and bending stresses:
1. An eccentric axial load P is equivalent to a centroidal axial load P plus a moment Pe.
2. For a rectangular section under uniaxial eccentric compression, e = d/6 is the no-tension boundary.
3. The kern of a solid circular section has radius D/8, where D is the section diameter.
4. If an eccentric compressive load lies outside the kern, the linear elastic stress distribution predicts tensile stress over part of the section.
Which statements are correct?
1. An eccentric axial load P is equivalent to a centroidal axial load P plus a moment Pe.
2. For a rectangular section under uniaxial eccentric compression, e = d/6 is the no-tension boundary.
3. The kern of a solid circular section has radius D/8, where D is the section diameter.
4. If an eccentric compressive load lies outside the kern, the linear elastic stress distribution predicts tensile stress over part of the section.
Which statements are correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Statement 1 is the standard static equivalence for an eccentric axial force. Statement 2 follows from setting the minimum edge stress of a rectangle equal to zero. Statement 3 follows from Z/A for a solid circle and gives the circular kern radius D/8. Statement 4 is also correct because exceeding the kern makes the direct stress insufficient to offset bending tension at the far side. Therefore all four statements are correct and Option D is the unique answer.
వివరణ:
Question 25
ప్రశ్న 25
Match each combined-stress concept with its correct relationship:
1. Eccentric moment
2. Rectangular one-axis kern limit
3. Solid circular kern radius
4. General one-direction kern distance
P. D/8
Q. Pe
R. I/(Ac) = Z/A
S. d/6
Which matching is correct?
1. Eccentric moment
2. Rectangular one-axis kern limit
3. Solid circular kern radius
4. General one-direction kern distance
P. D/8
Q. Pe
R. I/(Ac) = Z/A
S. d/6
Which matching is correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
An axial force acting at eccentricity e produces the bending moment Pe, so item 1 matches Q. For a rectangular section, the limiting eccentricity in one principal direction is d/6, so item 2 matches S. A solid circular section has kern radius D/8, making item 3 match P. The general kern distance to avoid tension is I/(Ac), which is also Z/A, so item 4 matches R. Therefore Option B gives the correct complete matching.
వివరణ:
Answer Key సమాధానాల పట్టిక
-
Question 1 ప్రశ్న 1Answer: B. σ = P/A ± My/I, with the sign of the bending term determined by the fibre considered సమాధానం: B.
-
Question 2 ప్రశ్న 2Answer: C. A centroidal force P together with a bending moment M = Pe సమాధానం: C.
-
Question 3 ప్రశ్న 3Answer: A. σmax,min = (P/A)(1 ± 6e/d) సమాధానం: A.
-
Question 4 ప్రశ్న 4Answer: D. A uniform compressive stress P/A సమాధానం: D.
-
Question 5 ప్రశ్న 5Answer: B. 9 MPa compression and 1 MPa compression సమాధానం: B.
-
Question 6 ప్రశ్న 6Answer: C. 20 mm సమాధానం: C.
-
Question 7 ప్రశ్న 7Answer: A. d/6 సమాధానం: A.
-
Question 8 ప్రశ్న 8Answer: B. The load line must remain within a central zone of total width d/3, corresponding to |e| <= d/6. సమాధానం: B.
-
Question 9 ప్రశ్న 9Answer: D. |ex|/(b/6) + |ey|/(d/6) <= 1 సమాధానం: D.
-
Question 10 ప్రశ్న 10Answer: C. D/8 సమాధానం: C.
-
Question 11 ప్రశ్న 11Answer: A. 22.92 MPa compression and 2.55 MPa compression సమాధానం: A.
-
Question 12 ప్రశ్న 12Answer: D. 750 kN సమాధానం: D.
-
Question 13 ప్రశ్న 13Answer: B. 25 MPa tension and 5 MPa compression సమాధానం: B.
-
Question 14 ప్రశ్న 14Answer: C. A linear triangular distribution, zero at one edge and maximum compression at the opposite edge సమాధానం: C.
-
Question 15 ప్రశ్న 15Answer: A. d/3 సమాధానం: A.
-
Question 16 ప్రశ్న 16Answer: D. 33.3 mm సమాధానం: D.
-
Question 17 ప్రశ్న 17Answer: B. 13.26 MPa compression and zero stress సమాధానం: B.
-
Question 18 ప్రశ్న 18Answer: C. 8.5 MPa compression and 1.5 MPa compression సమాధానం: C.
-
Question 19 ప్రశ్న 19Answer: A. 11 MPa compression and 1 MPa tension సమాధానం: A.
-
Question 20 ప్రశ్న 20Answer: D. k = I/(Ac) = Z/A సమాధానం: D.
-
Question 21 ప్రశ్న 21Answer: B. 18 MPa compression and 2 MPa compression సమాధానం: B.
-
Question 22 ప్రశ్న 22Answer: C. 22 MPa compression and 2 MPa tension సమాధానం: C.
-
Question 23 ప్రశ్న 23Answer: A. 15 mm సమాధానం: A.
-
Question 24 ప్రశ్న 24Answer: D. 1, 2, 3 and 4 సమాధానం: D.
-
Question 25 ప్రశ్న 25Answer: B. 1-Q, 2-S, 3-P, 4-R సమాధానం: B.