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Question 1
ప్రశ్న 1
For a stable linear-elastic structure whose strain energy U is expressed in terms of an applied force P, which relation gives the displacement in the direction of P according to Castigliano's theorem?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Castigliano's theorem relates generalized displacement to the derivative of elastic strain energy with respect to the corresponding generalized load. For a force P, the conjugate displacement is therefore obtained as ∂U/∂P. The theorem is especially convenient when the internal force resultants can be written in terms of the applied loads. In linear-elastic systems, strain energy and complementary energy lead to the familiar displacement form used in structural analysis. Therefore Option A is correct.
వివరణ:
Question 2
ప్రశ్న 2
A moment M is applied to a linear-elastic structure, and the strain energy is written as U(M). Which quantity is obtained from ∂U/∂M?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A moment is a generalized force whose energetically conjugate generalized displacement is a rotation. Castigliano's theorem therefore gives the corresponding rotation by differentiating strain energy with respect to that moment. The same principle applies to a force and its associated translation. The derivative must be taken while other independent generalized loads are treated appropriately. Hence Option A is correct.
వివరణ:
Question 3
ప్రశ్న 3
For a beam in which bending deformation dominates, the real-load bending moment is M(x) and a unit load applied in the direction of the required displacement produces moment m(x). Which unit-load expression gives the displacement?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The unit-load method is obtained from the principle of virtual work. The real structure develops moment M(x), while the virtual unit load develops moment m(x). Their product divided by flexural rigidity gives the elemental virtual-work contribution associated with bending. Integrating this quantity over all beam segments gives the requested displacement. Therefore Option A is correct.
వివరణ:
Question 4
ప్రశ్న 4
For a pin-jointed truss in which only axial member deformation is considered, which expression gives a displacement by the unit-load method?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Ni denotes the axial force in member i due to the real external loading. The quantity ni denotes the member force produced by a unit load acting in the direction of the required displacement. The virtual-work contribution of each member is Ni ni Li/(AiEi). Summation over all truss members gives the desired signed displacement. Therefore Option A is correct.
వివరణ:
Question 5
ప్రశ్న 5
A displacement is required at a point where no actual external load is applied. How is Castigliano's theorem conveniently used?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Castigliano's theorem requires differentiation with respect to a generalized load associated with the desired generalized displacement. If no real load already acts at that coordinate, an auxiliary or dummy load can be introduced. The internal actions and strain energy are then written in terms of that dummy load and the actual loads. After differentiation with respect to the dummy load, the dummy load is set to its actual value, which is zero. Hence Option A is correct.
వివరణ:
Question 6
ప్రశ్న 6
Which set of assumptions is most consistent with the usual elementary use of Castigliano and unit-load methods for structural displacement?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Elementary energy methods are normally developed for stable structures undergoing small elastic deformations. Linear behaviour permits superposition and makes the usual strain-energy expressions directly applicable. The deformation mode included in the calculation must also be consistent with the structural idealization. These methods can be applied to both determinate and indeterminate structures when properly formulated. Therefore Option A is correct.
వివరణ:
Question 7
ప్రశ్న 7
A cantilever beam is 2.4 m long and carries a 9 kN downward point load at its free end. If E = 200 GPa and I = 72 × 10^6 mm^4, what is the free-end vertical deflection obtained from the bending strain-energy method?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For a cantilever with a free-end point load, the bending strain energy is obtained from M = P(L-x). Differentiating the integrated energy with respect to P gives δ = PL³/(3EI). Substitution of P = 9000 N, L = 2400 mm, E = 200000 N/mm² and I = 72 × 10^6 mm^4 gives 2.88 mm. The displacement is in the direction of the applied load. Therefore Option C is correct.
వివరణ:
Question 8
ప్రశ్న 8
For the cantilever in the previous question, what is the free-end rotation due to the 9 kN end load?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The free-end rotation of a prismatic cantilever under an end point load is PL²/(2EI). This result can be obtained by introducing a dummy end moment and differentiating the bending strain energy. Substitution gives 9000 × 2400² divided by 2 × 200000 × 72 × 10^6. The result is 0.0018 rad. Therefore Option C is correct.
వివరణ:
Question 9
ప్రశ్న 9
A simply supported beam of span 3.6 m carries an 18 kN central point load. If E = 200 GPa and I = 60 × 10^6 mm^4, what is the midspan deflection by an energy or unit-load method?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For a simply supported beam with a central point load, the midspan deflection is PL³/(48EI). This expression follows directly from either Castigliano's theorem or the unit-load bending integral. Substituting the stated values gives 1.458 mm. The displacement is downward because the load and resulting curvature produce downward flexure at midspan. Therefore Option C is correct.
వివరణ:
Question 10
ప్రశ్న 10
A simply supported beam of span 3.6 m carries a uniformly distributed load of 5 kN/m over the entire span. If E = 200 GPa and I = 90 × 10^6 mm^4, what is the midspan deflection approximately?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The midspan deflection of a simply supported prismatic beam under full-span UDL is 5wL⁴/(384EI). The load intensity 5 kN/m is numerically 5 N/mm. Substituting L = 3600 mm, E = 200000 N/mm² and I = 90 × 10^6 mm^4 gives 0.6075 mm. This is approximately 0.608 mm downward. Therefore Option B is correct.
వివరణ:
Question 11
ప్రశ్న 11
A simply supported beam has span 5 m. A 24 kN point load acts 2 m from the left support and 3 m from the right support. If E = 200 GPa and I = 100 × 10^6 mm^4, what is the vertical deflection directly under the load?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For a simply supported beam carrying a point load P at distances a and b from the supports, the deflection at the load is Pa²b²/(3EIL). Here a = 2000 mm, b = 3000 mm and L = 5000 mm. Substitution of the stated data gives 2.88 mm. An equivalent result follows by integrating the product of the real and unit-load moment diagrams. Therefore Option C is correct.
వివరణ:
Question 12
ప్రశ్న 12
A cantilever beam 2.4 m long is subjected only to a constant end moment of 8 kN·m. If E = 200 GPa and I = 72 × 10^6 mm^4, what is the rotation at the free end?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A constant end moment produces constant bending moment along the cantilever. The bending strain energy is M²L/(2EI). Differentiating this energy with respect to M gives the end rotation ML/(EI). Substitution yields 0.001333 rad. Therefore Option C is correct.
వివరణ:
Question 13
ప్రశ్న 13
For the cantilever in the previous question, what is the transverse displacement of the free end caused by the 8 kN·m end moment?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The required free-end translation can be obtained by introducing a dummy transverse force at the free end. Castigliano differentiation or the unit-load method gives δ = ML²/(2EI). Substituting M = 8 × 10^6 N·mm and L = 2400 mm gives 1.60 mm. The direction depends on the sense of the applied end moment. Therefore Option C is correct.
వివరణ:
Question 14
ప్రశ్న 14
A cantilever beam of length 2 m has E = 200 GPa and I = 80 × 10^6 mm^4. It carries an 8 kN downward point load at the free end and a 6 kN·m end moment acting in the same sense as the bending produced by that load. What is the free-end vertical deflection?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Because the structure is linear elastic, the displacement contributions from the end force and end moment may be superposed. The load contribution is PL³/(3EI), which is approximately 1.333 mm. The moment contribution is ML²/(2EI), which is 0.750 mm. Since the two actions bend the cantilever in the same sense, the contributions add to 2.083 mm. Therefore Option C is correct.
వివరణ:
Question 15
ప్రశ్న 15
In the unit-load method, what virtual action should be applied to determine the rotation at a specified point of a beam?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The virtual load must be energetically conjugate to the displacement quantity being sought. A rotation is conjugate to a moment. Therefore a unit moment is applied at the point and in the selected positive rotational sense. The resulting virtual bending-moment diagram is then combined with the real bending-moment diagram. Hence Option A is correct.
వివరణ:
Question 16
ప్రశ్న 16
A unit-load calculation for a vertical displacement gives δ = -3.2 mm when the unit load was applied downward. What does the negative sign mean?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The unit-load direction establishes the positive direction for the calculated displacement. A negative result means the actual displacement is opposite to that selected positive direction. Since the virtual unit load was downward, the negative result indicates upward movement. The strain energy itself need not be negative because this sign arises from the real-virtual work product. Therefore Option A is correct.
వివరణ:
Question 17
ప్రశ్న 17
A steel bar 2 m long and 800 mm² in area carries a 40 kN axial tensile load. If E = 200 GPa, what axial displacement is obtained by differentiating U = P²L/(2AE) with respect to P?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Differentiating U = P²L/(2AE) with respect to P gives δ = PL/(AE). Substitution gives 40000 × 2000 divided by 800 × 200000. The result is 0.50 mm. This agrees with the ordinary axial-deformation relation and illustrates Castigliano's theorem directly. Therefore Option B is correct.
వివరణ:
Question 18
ప్రశ్న 18
A circular shaft carries a torque of 1.2 kN·m over a length of 1.5 m. If G = 80 GPa and J = 2.0 × 10^6 mm^4, what rotation is obtained by differentiating the torsional strain energy with respect to torque?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The torsional strain energy of the prismatic shaft is U = T²L/(2GJ). Differentiating with respect to T gives θ = TL/(GJ). Substitution of T = 1.2 × 10^6 N·mm and L = 1500 mm gives 0.01125 rad. This is the familiar elastic angle-of-twist result. Therefore Option B is correct.
వివరణ:
Question 19
ప్రశ్న 19
A symmetric triangular truss has joints A(0,0), B(6,0) and C(3,4) in metres, with members AB, AC and BC. A is pinned, B is a vertical roller, and a 24 kN downward load acts at C. All members have A = 800 mm² and E = 200 GPa. Neglecting bending and shear deformation, what is the downward displacement of C?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The two inclined members are 3-4-5 members and each carries 15 kN compression under the real load. The bottom member AB carries 9 kN tension. Under a unit downward load at C, the virtual forces are -0.625 in each diagonal and +0.375 in AB when tension is positive. Summation of Ni ni Li/(AE) for the three members gives 0.7125 mm. Therefore Option C is correct.
వివరణ:
Question 20
ప్రశ్న 20
For the same triangular truss A(0,0), B(6,0), C(3,4) m with A pinned and B a vertical roller, a 24 kN downward load acts at C and all members have A = 800 mm² and E = 200 GPa. What is the horizontal displacement of joint B?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Joint A has no horizontal displacement because it is pinned to the support. Member AB is horizontal and carries 9 kN tension under the actual loading. Its axial extension is therefore NL/(AE) = 9000 × 6000/(800 × 200000). This equals 0.3375 mm, and the roller at B permits that horizontal movement. Therefore joint B moves approximately 0.338 mm to the right and Option B is correct.
వివరణ:
Question 21
ప్రశ్న 21
A triangular truss has joints A(0,0), B(4,0) and C(1,3) in metres, members AB, AC and BC, a pin at A and a vertical roller at B. A 12 kN downward load acts at C. All members have A = 600 mm² and E = 200 GPa. What is the downward displacement of C approximately?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The support reactions are Ay = 9 kN and By = 3 kN. The member forces are AB = 3 kN tension, AC = 3√10 kN compression and BC = 3√2 kN compression. Because the virtual load is a unit downward load applied at the same point and direction as the single real load, the virtual member forces are the corresponding real member forces divided by 12 kN. Substitution in ΣNi niLi/(AE) gives approximately 0.2757 mm downward. Therefore Option B is correct.
వివరణ:
Question 22
ప్రశ్న 22
For a statically determinate truss subjected to fixed external loads, all member areas are doubled while E, geometry and supports remain unchanged. If axial deformation is the only displacement source, how does a displacement calculated by the unit-load method change?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
In a statically determinate truss, the real and virtual member forces are governed by equilibrium and are independent of member stiffness. The displacement contribution of each member is Ni ni Li/(AiEi). Doubling every area Ai therefore halves every member contribution. The total displacement consequently becomes one-half of its original value. Therefore Option B is correct.
వివరణ:
Question 23
ప్రశ్న 23
Which statement correctly compares Castigliano's theorem with the unit-load method for linear-elastic structural displacement calculations?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Both methods are based on work and energy principles for compatible structural response. In a linear-elastic problem, Castigliano differentiation and the corresponding unit-load calculation lead to the same physical displacement. Either method may be used for beams, trusses or frames when the appropriate internal deformation terms are included. Unit moments can also be used to obtain rotations by virtual work. Therefore Option A is correct.
వివరణ:
Question 24
ప్రశ్న 24
Consider the following statements:
1. In a beam-bending unit-load calculation, the desired displacement is obtained from an integral involving the product of the real and virtual bending moments.
2. In a truss, tension and compression signs must be retained when evaluating ΣNi niLi/(AiEi).
3. Castigliano differentiation with respect to an applied moment gives the conjugate rotation.
4. A negative unit-load result always means the strain energy of the structure is negative.
Which statements are correct?
1. In a beam-bending unit-load calculation, the desired displacement is obtained from an integral involving the product of the real and virtual bending moments.
2. In a truss, tension and compression signs must be retained when evaluating ΣNi niLi/(AiEi).
3. Castigliano differentiation with respect to an applied moment gives the conjugate rotation.
4. A negative unit-load result always means the strain energy of the structure is negative.
Which statements are correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Statement 1 is correct because beam displacement by virtual work contains the real-virtual moment product divided by EI. Statement 2 is correct because opposite signs of real and virtual truss forces can produce negative displacement contributions. Statement 3 is correct because a moment is conjugate to a rotation in Castigliano's theorem. Statement 4 is incorrect because a negative displacement merely indicates direction relative to the chosen virtual-load direction. Therefore Statements 1, 2 and 3 are correct and Option B is the unique answer.
వివరణ:
Question 25
ప్రశ్న 25
Match each structural-deflection concept with its correct expression or procedure:
1. Beam displacement due to bending by unit load
2. Truss displacement due to axial deformation
3. Castigliano displacement corresponding to force P
4. Rotation at a point by virtual work
P. ∂U/∂P
Q. Apply a unit moment at the required rotational coordinate
R. ∫Mm/(EI) dx
S. ΣNnL/(AE)
Which matching is correct?
1. Beam displacement due to bending by unit load
2. Truss displacement due to axial deformation
3. Castigliano displacement corresponding to force P
4. Rotation at a point by virtual work
P. ∂U/∂P
Q. Apply a unit moment at the required rotational coordinate
R. ∫Mm/(EI) dx
S. ΣNnL/(AE)
Which matching is correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Beam displacement due to bending is obtained from the integral of the real and virtual moment product divided by EI, so item 1 matches R. Truss displacement from member axial deformation is ΣNnL/(AE), so item 2 matches S. Castigliano's theorem gives the displacement corresponding to P as ∂U/∂P, so item 3 matches P. A rotational displacement is obtained in virtual work by applying a unit moment in the required rotational direction, so item 4 matches Q. Therefore Option A gives the correct complete matching.
వివరణ:
Answer Key సమాధానాల పట్టిక
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Question 1 ప్రశ్న 1Answer: A. δ = ∂U/∂P సమాధానం: A.
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Question 2 ప్రశ్న 2Answer: A. Rotation corresponding to M సమాధానం: A.
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Question 3 ప్రశ్న 3Answer: A. δ = ∫ M(x)m(x)/(EI) dx సమాధానం: A.
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Question 4 ప్రశ్న 4Answer: A. δ = Σ(Ni ni Li/AiEi) సమాధానం: A.
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Question 5 ప్రశ్న 5Answer: A. Introduce a dummy load Q in the desired direction, express U including Q, evaluate ∂U/∂Q, and then set Q = 0. సమాధానం: A.
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Question 6 ప్రశ్న 6Answer: A. Stable structure, small displacements, linear-elastic response and compatible deformation assumptions సమాధానం: A.
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Question 7 ప్రశ్న 7Answer: C. 2.88 mm downward సమాధానం: C.
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Question 8 ప్రశ్న 8Answer: C. 0.0018 rad సమాధానం: C.
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Question 9 ప్రశ్న 9Answer: C. 1.458 mm downward సమాధానం: C.
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Question 10 ప్రశ్న 10Answer: B. 0.608 mm సమాధానం: B.
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Question 11 ప్రశ్న 11Answer: C. 2.88 mm సమాధానం: C.
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Question 12 ప్రశ్న 12Answer: C. 0.001333 rad సమాధానం: C.
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Question 13 ప్రశ్న 13Answer: C. 1.60 mm సమాధానం: C.
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Question 14 ప్రశ్న 14Answer: C. 2.083 mm downward సమాధానం: C.
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Question 15 ప్రశ్న 15Answer: A. A unit moment at that point in the selected positive rotational direction సమాధానం: A.
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Question 16 ప్రశ్న 16Answer: A. The actual displacement is 3.2 mm upward. సమాధానం: A.
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Question 17 ప్రశ్న 17Answer: B. 0.50 mm సమాధానం: B.
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Question 18 ప్రశ్న 18Answer: B. 0.01125 rad సమాధానం: B.
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Question 19 ప్రశ్న 19Answer: C. 0.713 mm సమాధానం: C.
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Question 20 ప్రశ్న 20Answer: B. 0.338 mm to the right సమాధానం: B.
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Question 21 ప్రశ్న 21Answer: B. 0.276 mm సమాధానం: B.
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Question 22 ప్రశ్న 22Answer: B. It becomes one-half. సమాధానం: B.
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Question 23 ప్రశ్న 23Answer: A. They are consistent energy or virtual-work approaches and should give the same displacement when the same deformation effects are included. సమాధానం: A.
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Question 24 ప్రశ్న 24Answer: B. 1, 2 and 3 only సమాధానం: B.
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Question 25 ప్రశ్న 25Answer: A. 1-R, 2-S, 3-P, 4-Q సమాధానం: A.