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Question 1
ప్రశ్న 1
Which statement most appropriately distinguishes a column from a strut in elementary structural terminology?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Both columns and struts are structural members that primarily carry axial compression. The term column is commonly used for compression members that are predominantly vertical. The term strut is broader and may include inclined or horizontal compression members. Either type can fail by buckling when sufficiently slender. Therefore Option A gives the appropriate elementary distinction.
వివరణ:
Question 2
ప్రశ్న 2
Which set of assumptions is consistent with the classical Euler theory for an ideal long column?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Euler theory idealizes the column as initially perfectly straight and prismatic. The material is assumed homogeneous and linearly elastic, so Hooke's law remains applicable at buckling. The compressive load is assumed to act concentrically through the centroidal axis without initial eccentricity. The member must also be sufficiently slender for elastic instability rather than crushing or inelastic buckling to govern. Therefore Option A correctly states the classical assumptions.
వివరణ:
Question 3
ప్రశ్న 3
For an ideal elastic column having flexural rigidity EI and effective length Le, which expression gives the lowest Euler critical load?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Euler's lowest elastic buckling load is inversely proportional to the square of the effective column length. It is directly proportional to Young's modulus and to the second moment of area about the relevant buckling axis. The standard expression is Pcr = π²EI/Le². End conditions are incorporated through the effective length rather than by changing the basic form of the equation. Therefore Option A is correct.
వివరణ:
Question 4
ప్రశ్న 4
Which set of effective-length factors K is correct for the four standard ideal column end conditions, where Le = KL?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A pin-ended column has effective length equal to its actual unsupported length, so K = 1.0. A column fixed at both ends has an ideal effective length L/2, giving K = 0.5. A cantilever column fixed at one end and free at the other has effective length 2L, so K = 2.0. The ideal fixed-pinned case has K approximately equal to 0.699, commonly rounded to 0.7. Therefore Option A is correct.
వివరణ:
Question 5
ప్రశ్న 5
Two ideal columns have identical E, I and actual length L. Column A is pinned at both ends and Column B is fixed at both ends. According to Euler theory, what is Pcr,B/Pcr,A?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The effective length of the pin-ended column is L. The effective length of the fixed-fixed column is L/2. Since Euler load varies inversely with the square of effective length, the ratio is L²/(L/2)². This gives a critical-load ratio of 4. Therefore Option D is correct.
వివరణ:
Question 6
ప్రశ్న 6
Two ideal columns have the same E, I and actual length. One is pinned at both ends and the other is fixed at one end and free at the other. What is the ratio of the cantilever Euler load to the pin-ended Euler load?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The pin-ended column has effective length L. The fixed-free column has effective length 2L. Euler load is inversely proportional to the square of effective length. Therefore the ratio is L²/(2L)² = 1/4. Hence Option A is correct.
వివరణ:
Question 7
ప్రశ్న 7
An ideal fixed-pinned column and an ideal pinned-pinned column have identical E, I and actual length L. Using K = 0.699 for the fixed-pinned column, approximately how many times larger is its Euler critical load?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The pinned-pinned column has effective length L. The fixed-pinned column has effective length 0.699L. The critical-load ratio is therefore 1/(0.699²). This evaluates to approximately 2.05. Thus the ideal fixed-pinned restraint approximately doubles the Euler load and Option B is correct.
వివరణ:
Question 8
ప్రశ్న 8
Which pair correctly defines radius of gyration r and effective slenderness ratio λ for a column cross-section of area A and second moment I about the considered buckling axis?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Radius of gyration is defined from I = Ar². Therefore r = sqrt(I/A). Column slenderness must incorporate the effective length appropriate to the end restraints. The effective slenderness ratio is consequently λ = Le/r. Hence Option B is correct.
వివరణ:
Question 9
ప్రశ్న 9
A column cross-section has unequal principal moments of inertia Ix and Iy, with Iy < Ix. If all other conditions are unchanged, about which axis will an ideal Euler column normally buckle first?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Euler critical load is directly proportional to the second moment of area about the buckling axis. The smaller moment of inertia therefore produces the smaller critical load. Buckling occurs first in the mode requiring the least load. Equivalently, this is the direction associated with the minimum radius of gyration and maximum slenderness ratio. Therefore Option B is correct.
వివరణ:
Question 10
ప్రశ్న 10
A pin-ended steel column is 3.0 m long and has E = 200 GPa and I = 8.0 × 10^6 mm^4 about its governing buckling axis. What is its ideal Euler critical load approximately?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For a pin-ended column, the effective length equals the actual length of 3000 mm. Euler's equation gives Pcr = π²EI/L². Substituting E = 200000 N/mm² and I = 8.0 × 10^6 mm^4 gives approximately 1.7546 × 10^6 N. This is 1754.6 kN. Therefore Option C is correct.
వివరణ:
Question 11
ప్రశ్న 11
The same ideal column has E = 200 GPa, I = 8.0 × 10^6 mm^4 and actual length 3.0 m, but both ends are perfectly fixed. What Euler critical load is predicted?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For fixed-fixed ends, the effective length is L/2 = 1500 mm. Euler load varies inversely with effective length squared. The fixed-fixed load is therefore four times the pin-ended value for the same E, I and actual length. Four times 1754.6 kN is approximately 7018.4 kN. Therefore Option C is correct.
వివరణ:
Question 12
ప్రశ్న 12
The same ideal 3.0 m column with E = 200 GPa and I = 8.0 × 10^6 mm^4 is fixed at one end and free at the other. What is its Euler critical load approximately?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A fixed-free column has effective length 2L, which is 6000 mm in this case. Euler load is therefore one-quarter of the corresponding pin-ended value. One-quarter of 1754.6 kN is approximately 438.6 kN. The reduced load reflects the much weaker buckling restraint of the cantilever condition. Therefore Option B is correct.
వివరణ:
Question 13
ప్రశ్న 13
An ideal column has E = 200 GPa, I = 8.0 × 10^6 mm^4, actual length 3.0 m, one end fixed and the other pinned. Using K = 0.699, what is the Euler critical load approximately?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The effective length is 0.699 × 3000 = 2097 mm. Euler's formula is then applied using this effective length. Substitution of E = 200000 N/mm² and I = 8.0 × 10^6 mm^4 gives approximately 3.591 × 10^6 N. This equals about 3591.1 kN. Therefore Option C is correct.
వివరణ:
Question 14
ప్రశ్న 14
A pin-ended rectangular steel column has a 100 mm by 200 mm cross-section, length 3.0 m and E = 200 GPa. What is the governing ideal Euler critical load?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The column buckles about the weaker centroidal axis. For the 100 mm by 200 mm rectangle, the smaller second moment is 200 × 100³/12 = 16.667 × 10^6 mm^4. With pinned ends, the effective length is 3000 mm. Euler's formula then gives approximately 3.6554 × 10^6 N. Therefore the governing ideal critical load is 3655.4 kN and Option B is correct.
వివరణ:
Question 15
ప్రశ్న 15
A pin-ended solid circular steel column has diameter 80 mm, length 2.0 m and E = 200 GPa. Which pair correctly gives its slenderness ratio and ideal Euler critical load?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For a solid circular section, the radius of gyration is d/4. Thus r = 80/4 = 20 mm. The pin-ended slenderness ratio is 2000/20 = 100. Using I = πd^4/64 in Euler's equation gives a critical load of approximately 992.2 kN. Therefore Option C is correct.
వివరణ:
Question 16
ప్రశ్న 16
An ideal elastic column has E = 200 GPa and effective slenderness ratio Le/r = 120. What is its Euler critical stress approximately?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Euler critical stress can be written as σcr = π²E/λ². Here E = 200000 MPa and λ = 120. Substitution gives π² × 200000/120². The result is approximately 137.08 MPa. Therefore Option B is correct.
వివరణ:
Question 17
ప్రశ్న 17
An ideal pin-ended column is modified so that its actual length is doubled while E, I and end conditions remain unchanged. How does the Euler critical load change?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For unchanged end conditions, effective length is directly proportional to actual length. Euler load varies inversely as the square of effective length. Doubling the length therefore multiplies the denominator by four. The critical load consequently becomes one-quarter of its original value. Hence Option C is correct.
వివరణ:
Question 18
ప్రశ్న 18
A solid circular Euler column is redesigned with its diameter increased by 20 percent while its length, material and end restraints remain unchanged. By what factor does its ideal Euler critical load increase?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Euler load is proportional to the second moment of area I. For a solid circular section, I is proportional to d^4. Increasing diameter by 20 percent therefore multiplies I by 1.2^4. This factor is 2.0736. Hence the ideal Euler load increases by the same factor and Option D is correct.
వివరణ:
Question 19
ప్రశ్న 19
A pin-ended column 2.5 m long is required to have an ideal Euler critical load of 800 kN. If E = 200 GPa, what minimum second moment of area is required about the governing buckling axis?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For pin-ended conditions, the effective length is 2500 mm. Rearranging Euler's equation gives I = PcrL²/(π²E). Substitution of 800000 N, 2500 mm and 200000 N/mm² gives approximately 2.533 × 10^6 mm^4. This is the theoretical minimum inertia required by the ideal Euler criterion. Therefore Option B is correct.
వివరణ:
Question 20
ప్రశ్న 20
A pin-ended solid circular steel column is 2.0 m long and has E = 200 GPa. What theoretical diameter is required for an ideal Euler load of 500 kN?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The required second moment is obtained from I = PcrL²/(π²E). This gives approximately 1.0132 × 10^6 mm^4. For a solid circle, I = πd^4/64. Solving this relation gives d approximately equal to 67.4 mm. Therefore Option C is correct.
వివరణ:
Question 21
ప్రశ్న 21
A column is 1.5 m long, fixed at one end and free at the other, and has a minimum radius of gyration of 15 mm. What is its effective slenderness ratio?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A fixed-free column has effective length equal to twice its actual length. The effective length is therefore 2 × 1500 = 3000 mm. Slenderness ratio is Le/r. Dividing 3000 mm by 15 mm gives 200. Therefore Option D is correct.
వివరణ:
Question 22
ప్రశ్న 22
Which statement best describes the effective length of an ideal Euler column?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Effective length is a convenient way of representing the influence of end restraint on elastic buckling. It is defined so that an equivalent pin-ended column of that length has the same Euler critical load. For example, a fixed-fixed ideal column of physical length L has effective length L/2. A fixed-free column has effective length 2L. Therefore Option B is correct.
వివరణ:
Question 23
ప్రశ్న 23
A material has E = 200 GPa and proportional-limit stress of 250 MPa. According to the ideal Euler stress relation, approximately what effective slenderness ratio makes the Euler critical stress equal to the proportional limit?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Euler theory requires the predicted buckling stress to remain within the elastic range for its classical material assumption to be valid. Setting π²E/λ² equal to 250 MPa gives λ = πsqrt(E/250). With E = 200000 MPa, this gives approximately 88.9. A substantially smaller slenderness ratio would make the Euler prediction exceed the proportional limit and would indicate intermediate or inelastic behaviour rather than a valid classical Euler case. Therefore Option C is correct.
వివరణ:
Question 24
ప్రశ్న 24
For an ideal pin-ended Euler column, the lowest buckling mode has critical load P1. What is the elastic eigenvalue associated with a mode containing n half-sine waves along the length?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The Euler eigenvalues for an ideal pin-ended column are proportional to the square of the mode number. The lowest mode corresponds to n = 1 and governs the first loss of stability. A mode with n half-sine waves has critical load n² times the first-mode load. Higher modes therefore require larger ideal eigenvalues than the fundamental mode. Hence Option B is correct.
వివరణ:
Question 25
ప్రశ్న 25
Two pin-ended columns have the same material, cross-sectional area and length. Column A has minimum radius of gyration 30 mm, while Column B has minimum radius of gyration 20 mm. What is the ratio Pcr,A/Pcr,B according to Euler theory?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For equal area, the second moment of area can be written as I = Ar². Since the columns have the same E, A, effective length and end condition, Euler load is proportional to r². The ratio is therefore 30²/20². This equals 900/400 = 2.25. Therefore Column A has 2.25 times the ideal Euler load and Option C is correct.
వివరణ:
Answer Key సమాధానాల పట్టిక
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Question 1 ప్రశ్న 1Answer: A. A column is generally a compression member used predominantly in a vertical orientation, whereas a strut is a compression member that may have any orientation. సమాధానం: A.
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Question 2 ప్రశ్న 2Answer: A. The column is initially straight, homogeneous, linearly elastic, prismatic, centrally loaded and sufficiently slender, with idealized end restraints. సమాధానం: A.
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Question 3 ప్రశ్న 3Answer: A. Pcr = π²EI/Le² సమాధానం: A.
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Question 4 ప్రశ్న 4Answer: A. Pinned-pinned: 1.0; fixed-fixed: 0.5; fixed-free: 2.0; fixed-pinned: approximately 0.699 సమాధానం: A.
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Question 5 ప్రశ్న 5Answer: D. 4 సమాధానం: D.
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Question 6 ప్రశ్న 6Answer: A. 1/4 సమాధానం: A.
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Question 7 ప్రశ్న 7Answer: B. 2.05 times సమాధానం: B.
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Question 8 ప్రశ్న 8Answer: B. r = sqrt(I/A) and λ = Le/r సమాధానం: B.
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Question 9 ప్రశ్న 9Answer: B. About the axis having the smaller moment Iy సమాధానం: B.
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Question 10 ప్రశ్న 10Answer: C. 1754.6 kN సమాధానం: C.
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Question 11 ప్రశ్న 11Answer: C. 7018.4 kN సమాధానం: C.
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Question 12 ప్రశ్న 12Answer: B. 438.6 kN సమాధానం: B.
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Question 13 ప్రశ్న 13Answer: C. 3591.1 kN సమాధానం: C.
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Question 14 ప్రశ్న 14Answer: B. 3655.4 kN సమాధానం: B.
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Question 15 ప్రశ్న 15Answer: C. λ = 100 and Pcr ≈ 992.2 kN సమాధానం: C.
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Question 16 ప్రశ్న 16Answer: B. 137.08 MPa సమాధానం: B.
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Question 17 ప్రశ్న 17Answer: C. It becomes one-quarter. సమాధానం: C.
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Question 18 ప్రశ్న 18Answer: D. 2.0736 సమాధానం: D.
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Question 19 ప్రశ్న 19Answer: B. 2.533 × 10^6 mm^4 సమాధానం: B.
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Question 20 ప్రశ్న 20Answer: C. 67.4 mm సమాధానం: C.
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Question 21 ప్రశ్న 21Answer: D. 200 సమాధానం: D.
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Question 22 ప్రశ్న 22Answer: B. It is the length of an equivalent pin-ended column having the same lowest Euler buckling load as the actual restrained column. సమాధానం: B.
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Question 23 ప్రశ్న 23Answer: C. 88.9 సమాధానం: C.
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Question 24 ప్రశ్న 24Answer: B. Pn = n²P1 సమాధానం: B.
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Question 25 ప్రశ్న 25Answer: C. 2.25 సమాధానం: C.