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Question 1
ప్రశ్న 1
A pin-jointed truss member of length L, cross-sectional area A and Young's modulus E carries a constant axial force N within the linear-elastic range. Taking extension as positive, which expression gives its axial deformation?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The normal stress in a prismatic truss member is N/A. The corresponding axial strain in the linear-elastic range is N/(AE). Multiplying the strain by the member length L gives the axial deformation NL/(AE). A tensile force gives positive extension under the stated sign convention, while a compressive force gives shortening. Therefore Option A is correct.
వివరణ:
Question 2
ప్రశ్న 2
For a statically determinate pin-jointed truss in which member axial deformation is the only significant source of displacement, which unit-load expression gives the displacement δ in a specified direction?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Ni denotes the member force caused by the real loading system. The quantity ni denotes the member force caused by a unit load applied at the point and in the direction of the required displacement. Virtual work equates the external virtual work to the sum of internal virtual work contributions of the members. For axial truss members this produces the summation Σ(Ni ni Li/AiEi). Therefore Option A is correct.
వివరణ:
Question 3
ప్రశ్న 3
In the unit-load method for finding the horizontal displacement of a truss joint J, what must be done to obtain the virtual member forces ni?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The virtual system must correspond to the displacement component being sought. For a horizontal displacement at J, a unit force is therefore placed horizontally at J in the selected positive direction. The truss is analysed under this virtual loading to obtain the forces ni in all relevant members. These forces are then multiplied by the corresponding real-load forces Ni in the virtual-work summation. Hence Option A is correct.
వివరణ:
Question 4
ప్రశ్న 4
Which statement most accurately expresses Betti's reciprocal-work theorem for two load systems P and Q acting separately on the same linear-elastic structure?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Betti's theorem compares two distinct loading systems acting on the same elastic structure. The reciprocal work of the first load system through displacements generated by the second equals the reciprocal work of the second through displacements generated by the first. The two loading systems need not have equal magnitudes or act at the same positions. The result follows from linear-elastic structural response and reciprocal energy relations. Therefore Option A is correct.
వివరణ:
Question 5
ప్రశ్న 5
Maxwell's reciprocal-displacement theorem is a special consequence of Betti's theorem. Which statement correctly expresses it for unit loads applied at generalized coordinates i and j?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Let fij denote the displacement at coordinate i caused by a unit generalized load at coordinate j. Maxwell reciprocity gives fij = fji for a linear-elastic reciprocal structure. The coordinates may be at different joints or may even represent different compatible generalized displacement types when properly paired. The theorem is not restricted to statically determinate structures. Therefore Option A is correct.
వివరణ:
Question 6
ప్రశ్న 6
Which set of conditions is most consistent with the usual application of the Maxwell-Betti reciprocal theorem in elementary structural analysis?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The classical reciprocal theorem is used for structures whose force-displacement relations are linear and elastic. Small displacements permit the original structural geometry to be used for both loading systems. The structural configuration and support restraints must remain the same when the reciprocal responses are compared. Stability is also necessary so that the displacement solution is well defined. Therefore Option A is correct.
వివరణ:
Question 7
ప్రశ్న 7
A linear-elastic structure is represented by the flexibility relation {δ} = [f]{P}. What property of the flexibility matrix follows from Maxwell's reciprocal theorem?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The coefficient fij is the displacement at coordinate i due to a unit load at coordinate j. Maxwell reciprocity states that this displacement equals the displacement at coordinate j caused by a unit load at coordinate i. Thus fij = fji. Equality of corresponding off-diagonal coefficients means that the flexibility matrix is symmetric. Therefore Option A is correct.
వివరణ:
Question 8
ప్రశ్న 8
Load system P consists of 20 kN acting at generalized coordinate 1 and 10 kN acting at coordinate 2. Under a second load system Q, the corresponding displacements at coordinates 1 and 2 are 0.30 mm and 0.10 mm in the positive P directions. If Q consists of a single 14 kN force at coordinate 3, what displacement at coordinate 3 is produced by load system P in the direction of Q?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The reciprocal work of system P through the displacements caused by Q is 20(0.30) + 10(0.10). This equals 7 kN·mm. Betti's theorem requires the reciprocal work of Q through the displacement caused by P to have the same value. Therefore 14δ3 = 7 kN·mm. Hence δ3 = 0.50 mm and Option B is correct.
వివరణ:
Question 9
ప్రశ్న 9
A 12 kN force applied at coordinate A of a linear-elastic structure produces a displacement of 0.84 mm at coordinate B in a specified positive direction. What displacement at A will be produced by a 20 kN force applied at B in the reciprocal positive direction?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The flexibility coefficient from A to B is 0.84/12 = 0.07 mm/kN. Maxwell reciprocity requires the reciprocal flexibility coefficient from B to A to have the same value. A 20 kN load at B therefore produces a displacement at A equal to 20 × 0.07 mm. This gives 1.40 mm in the corresponding positive direction. Therefore Option C is correct.
వివరణ:
Question 10
ప్రశ్న 10
A unit load at generalized coordinate A produces a displacement of -0.25 mm at coordinate B, where the negative sign refers to the selected positive displacement direction at B. What displacement at A is produced by a unit load at B in its selected positive direction?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Reciprocity applies to signed generalized displacements, not merely to their magnitudes. The flexibility coefficient fBA is given as -0.25 mm per unit load. Maxwell's theorem requires fAB = fBA when the generalized positive directions are retained consistently. The reciprocal displacement therefore also has the value -0.25 mm. Hence Option B is correct.
వివరణ:
Question 11
ప్రశ్న 11
A triangular truss has joints A(0,0), B(4,0) and C(2,3) in metres, with members AB, AC and BC. A is pinned and B is a vertical roller. All members have area 500 mm² and E = 200 GPa. A 30 kN downward load acts at C. Neglecting all deformation except axial deformation of the members, what is the downward displacement of C approximately?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Under the 30 kN real load, AB carries 10 kN tension while AC and BC each carry about 18.028 kN compression. A unit downward load at C produces corresponding virtual member forces of 1/3 in AB and -√13/6 in each diagonal when the unit load magnitude is taken consistently. The diagonal lengths are √13 m and the bottom member length is 4 m. Substitution in ΣNi niLi/(AE) gives approximately 0.9145 mm downward. Therefore Option B is correct.
వివరణ:
Question 12
ప్రశ్న 12
For the truss A(0,0), B(4,0), C(2,3) m with members AB, AC and BC, A pinned and B a vertical roller, all members have A = 500 mm² and E = 200 GPa. A 30 kN downward load acts at C. What is the horizontal displacement of C, taking rightward displacement as positive?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The real-load member forces are 10 kN tension in AB and 18.028 kN compression in each diagonal. A unit horizontal load acting to the right at C produces virtual forces 0.5 in AB, +√13/4 in AC and -√13/4 in BC. The two diagonal contributions to ΣNi niLi/(AE) cancel because their real forces and lengths are equal while the virtual forces have opposite signs. The remaining AB contribution gives 10 × 0.5 × 4000/(500 × 200) = 0.200 mm. Therefore C moves 0.200 mm to the right and Option C is correct.
వివరణ:
Question 13
ప్రశ్న 13
For the same triangular truss A(0,0), B(4,0), C(2,3) m with A = 500 mm² and E = 200 GPa in every member, Maxwell reciprocity may be checked directly. If a 30 kN downward load at C produces 0.200 mm rightward displacement of C, what vertical displacement of C is produced by a 30 kN rightward load at C, taking downward as positive?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The first response defines the cross-flexibility between the downward and rightward generalized coordinates at C. Its value is 0.200/30 mm per kN. Maxwell reciprocity requires the reciprocal cross-flexibility to be identical. Applying the same 30 kN magnitude in the rightward direction therefore produces 0.200 mm displacement in the reciprocal downward coordinate. Thus Option C is correct.
వివరణ:
Question 14
ప్రశ్న 14
In the triangular truss A(0,0), B(4,0), C(2,3) m, all members have A = 500 mm² and E = 200 GPa. Under a 30 kN downward load at C, member AC carries approximately 18.028 kN compression. What is the axial deformation of member AC?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Member AC has length √13 m, which is approximately 3605.55 mm. Its axial force is -18.028 kN when tension is taken as positive. The axial deformation is NL/(AE). Using E = 200 kN/mm² and A = 500 mm² gives approximately -0.650 mm. The negative sign indicates shortening, so Option B is correct.
వివరణ:
Question 15
ప్రశ్న 15
For the same truss with A = 500 mm² and E = 200 GPa in all members, a 30 kN downward load at C produces a downward displacement of approximately 0.9145 mm. What total axial strain energy is stored in the truss?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For a linear-elastic structure loaded gradually from zero to the final load, the external work equals one-half of the final load multiplied by its corresponding displacement. Thus U = 0.5 × 30 kN × 0.9145 mm. Since one kN·mm equals one joule, the resulting energy is approximately 13.72 J. The same result would be obtained by summing Ni²Li/(2AiEi) over all members. Therefore Option B is correct.
వివరణ:
Question 16
ప్రశ్న 16
A triangular truss has A(0,0), B(6,0) and C(2,3) in metres, with members AB, AC and BC. A is pinned and B is a vertical roller. All members have area 600 mm² and E = 200 GPa. A 15 kN downward load acts at C. What is the downward displacement of C approximately?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For the 15 kN real load, the member forces are approximately 6.667 kN tension in AB, 12.019 kN compression in AC and 8.333 kN compression in BC. Under a unit downward load at C, the corresponding force ratios are 4/9 in AB, about -0.80123 in AC and -5/9 in BC. The member lengths are 6000 mm, √13 × 1000 mm and 5000 mm respectively. Substitution in the unit-load summation gives approximately 0.6304 mm downward. Therefore Option B is correct.
వివరణ:
Question 17
ప్రశ్న 17
For the symmetric triangular truss A(0,0), B(4,0), C(2,3) m under a 30 kN downward load at C, E = 200 GPa in all members. If the diagonal members AC and BC each have area 500 mm² but the area of bottom member AB is increased from 500 mm² to 1000 mm², what is the new downward displacement of C approximately?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The member forces do not change because the truss is statically determinate and the external loading and geometry are unchanged. Only the axial flexibility contribution of member AB is altered. Doubling the area of AB halves that member's contribution to the unit-load displacement sum. The two diagonal contributions remain unchanged, giving a recalculated downward displacement of approximately 0.848 mm. Therefore Option C is correct.
వివరణ:
Question 18
ప్రశ్న 18
In the virtual-work expression δ = Σ(Ni ni Li/AiEi), a particular member is in compression under the real loading but in tension under the unit-load system. What is the sign of that member's contribution if tension is consistently treated as positive?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The unit-load expression contains the algebraic product Ni ni rather than the squares of the two forces. A compressive real force is negative when tension is chosen positive. A tensile virtual force is positive under the same convention. Their product is therefore negative. This signed contribution is essential for obtaining the correct displacement direction, so Option B is correct.
వివరణ:
Question 19
ప్రశ్న 19
Load system P consists of forces P1 and P2. Load system Q consists of forces Q1 and Q2 acting on the same structure. Which equation represents Betti's theorem when u1(Q) and u2(Q) are displacements produced by Q at the points and directions of P1 and P2, and u1(P) and u2(P) are the corresponding displacements produced by P at the points and directions of Q1 and Q2?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Betti's theorem is an equality of reciprocal work rather than an equality of total force or total displacement. The forces of system P are multiplied by the compatible displacements generated by system Q at their points and directions. The forces of system Q are likewise multiplied by the corresponding displacements generated by system P. The algebraic sums of these two reciprocal-work quantities are equal. Therefore Option A is correct.
వివరణ:
Question 20
ప్రశ్న 20
A unit moment applied at rotational coordinate A of a linear-elastic structure causes a rotation of 0.004 rad at rotational coordinate B. What rotation at A is caused by a unit moment applied at B, with consistent positive rotational directions?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Moments and rotations form a conjugate generalized-force and generalized-displacement pair. Maxwell reciprocity is therefore applicable to rotational coordinates as well as translational coordinates. The flexibility coefficient giving rotation at B from a unit moment at A is 0.004 rad per unit moment. The reciprocal coefficient giving rotation at A from a unit moment at B must be equal. Hence the rotation is 0.004 rad and Option C is correct.
వివరణ:
Question 21
ప్రశ్న 21
A unit vertical force at point A produces a clockwise rotation θ at point B in a linear-elastic frame. Which reciprocal statement is correct when the positive directions are defined consistently?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Reciprocity is formulated in terms of generalized conjugate loads and displacements. A force is conjugate to a translation, while a moment is conjugate to a rotation. The cross-flexibility relating rotation at B to force at A equals the reciprocal flexibility relating translation at A to moment at B. Consistent positive directions must be retained when interpreting the sign. Therefore Option A is correct.
వివరణ:
Question 22
ప్రశ్న 22
Which situation would invalidate a direct use of the classical Maxwell reciprocal-displacement relation between two loading cases?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Classical Maxwell reciprocity depends on a linear reciprocal force-displacement relationship. Significant material nonlinearity or large geometric changes can make the tangent stiffness depend on the loading path or current state. In such a case the simple equality of the original flexibility coefficients cannot be assumed. Different joint locations or different coordinate directions do not themselves invalidate reciprocity. Therefore Option A is correct.
వివరణ:
Question 23
ప్రశ్న 23
The basic truss unit-load expression Σ(Ni ni Li/AiEi) is being used for a pin-jointed structure. Which statement is most accurate if significant support settlement, temperature change or lack-of-fit also occurs?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The standard summation represents the contribution from axial strain produced by member forces. Temperature change, fabrication error or support movement may create additional imposed deformations that are not represented by NiLi/(AiEi) alone. Their virtual-work contributions must therefore be incorporated when those effects are part of the required displacement. The underlying virtual-work and reciprocal principles can still remain valid for an appropriately formulated linear system. Therefore Option A is correct.
వివరణ:
Question 24
ప్రశ్న 24
Consider the following statements:
1. Maxwell reciprocity implies fij = fji for a linear-elastic structure.
2. The unit-load method for a truss uses products of real-load and virtual-load member forces.
3. For a statically determinate truss, changing a member area changes the member forces caused by a fixed external load even when geometry is unchanged.
4. Betti's theorem compares reciprocal work for two load systems on the same structure.
Which statements are correct?
1. Maxwell reciprocity implies fij = fji for a linear-elastic structure.
2. The unit-load method for a truss uses products of real-load and virtual-load member forces.
3. For a statically determinate truss, changing a member area changes the member forces caused by a fixed external load even when geometry is unchanged.
4. Betti's theorem compares reciprocal work for two load systems on the same structure.
Which statements are correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Statement 1 is correct because Maxwell's theorem gives symmetry of the flexibility coefficients. Statement 2 is correct because truss displacement by virtual work contains the product Ni ni for each member. Statement 3 is incorrect for an ordinary statically determinate truss because equilibrium determines the member forces independently of member stiffnesses, although the displacements do change. Statement 4 is the essential reciprocal-work statement of Betti's theorem. Therefore Statements 1, 2 and 4 are correct and Option B is the unique answer.
వివరణ:
Question 25
ప్రశ్న 25
Match each concept with its correct interpretation:
1. Ni in a truss unit-load calculation
2. ni in a truss unit-load calculation
3. Maxwell reciprocal theorem
4. Betti reciprocal-work theorem
P. Member force caused by the real loading
Q. Equality of reciprocal generalized displacements or flexibility coefficients
R. Member force caused by a unit load associated with the required displacement
S. Equality of reciprocal work of two loading systems
Which matching is correct?
1. Ni in a truss unit-load calculation
2. ni in a truss unit-load calculation
3. Maxwell reciprocal theorem
4. Betti reciprocal-work theorem
P. Member force caused by the real loading
Q. Equality of reciprocal generalized displacements or flexibility coefficients
R. Member force caused by a unit load associated with the required displacement
S. Equality of reciprocal work of two loading systems
Which matching is correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Ni represents the axial force produced in member i by the actual or real loading system, so item 1 matches P. The quantity ni is obtained by applying a unit load corresponding to the desired displacement, so item 2 matches R. Maxwell's theorem expresses reciprocal displacement or flexibility equality and therefore matches Q. Betti's theorem equates reciprocal work between two loading systems and therefore matches S. Hence Option A gives the correct complete matching.
వివరణ:
Answer Key సమాధానాల పట్టిక
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Question 1 ప్రశ్న 1Answer: A. Δ = NL/(AE) సమాధానం: A.
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Question 2 ప్రశ్న 2Answer: A. δ = Σ(Ni ni Li/AiEi) సమాధానం: A.
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Question 3 ప్రశ్న 3Answer: A. Apply a unit horizontal load at J in the chosen positive displacement direction and analyse the truss under that unit load. సమాధానం: A.
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Question 4 ప్రశ్న 4Answer: A. The work of load system P through the displacements caused by Q equals the work of load system Q through the displacements caused by P. సమాధానం: A.
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Question 5 ప్రశ్న 5Answer: A. The displacement at i due to a unit load at j equals the displacement at j due to a unit load at i, with consistent positive directions. సమాధానం: A.
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Question 6 ప్రశ్న 6Answer: A. Stable structure, linear-elastic behaviour, small-displacement response and unchanged support conditions between the two loading cases సమాధానం: A.
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Question 7 ప్రశ్న 7Answer: A. fij = fji, so the flexibility matrix is symmetric. సమాధానం: A.
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Question 8 ప్రశ్న 8Answer: B. 0.50 mm సమాధానం: B.
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Question 9 ప్రశ్న 9Answer: C. 1.400 mm సమాధానం: C.
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Question 10 ప్రశ్న 10Answer: B. -0.25 mm సమాధానం: B.
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Question 11 ప్రశ్న 11Answer: B. 0.915 mm సమాధానం: B.
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Question 12 ప్రశ్న 12Answer: C. 0.200 mm rightward సమాధానం: C.
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Question 13 ప్రశ్న 13Answer: C. 0.200 mm downward సమాధానం: C.
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Question 14 ప్రశ్న 14Answer: B. 0.650 mm shortening సమాధానం: B.
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Question 15 ప్రశ్న 15Answer: B. 13.72 J సమాధానం: B.
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Question 16 ప్రశ్న 16Answer: B. 0.630 mm సమాధానం: B.
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Question 17 ప్రశ్న 17Answer: C. 0.848 mm సమాధానం: C.
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Question 18 ప్రశ్న 18Answer: B. Negative because Ni and ni have opposite signs సమాధానం: B.
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Question 19 ప్రశ్న 19Answer: A. P1u1(Q) + P2u2(Q) = Q1u1(P) + Q2u2(P) సమాధానం: A.
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Question 20 ప్రశ్న 20Answer: C. 0.004 rad సమాధానం: C.
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Question 21 ప్రశ్న 21Answer: A. A unit moment at B in the positive rotational direction produces a vertical displacement at A numerically equal to θ. సమాధానం: A.
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Question 22 ప్రశ్న 22Answer: A. The structure develops significant geometric or material nonlinearity between the loading cases. సమాధానం: A.
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Question 23 ప్రశ్న 23Answer: A. Additional compatible virtual-work terms may be required because Σ(Ni ni Li/AiEi) alone represents only displacement associated with the included axial-force member deformations. సమాధానం: A.
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Question 24 ప్రశ్న 24Answer: B. 1, 2 and 4 only సమాధానం: B.
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Question 25 ప్రశ్న 25Answer: A. 1-P, 2-R, 3-Q, 4-S సమాధానం: A.