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Question 1
ప్రశ్న 1
What does the degree of static indeterminacy of a stable structure represent?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Static indeterminacy is associated with unknown forces rather than unknown displacements. A statically indeterminate structure contains more independent force unknowns than can be found using equilibrium equations alone. The excess independent force quantities are called redundants. Compatibility and force-deformation relations are required in addition to equilibrium to determine them. Therefore Option B is correct.
వివరణ:
Question 2
ప్రశ్న 2
What is meant by kinematic indeterminacy in structural analysis?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Kinematic indeterminacy concerns displacement unknowns rather than force unknowns. These displacement coordinates may include joint translations and joint rotations depending on the structural model. In displacement-based methods, they form the primary unknowns of the analysis. Support restraints and modeling assumptions determine how many such independent coordinates remain. Therefore Option C is correct.
వివరణ:
Question 3
ప్రశ్న 3
Why must structural stability be checked before a numerical static-indeterminacy formula is interpreted?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Simple counting formulas compare the numbers of force unknowns and available equilibrium equations. They do not by themselves guarantee that the members and supports are arranged in a stable geometric configuration. A structure may satisfy a determinacy count yet still possess a mechanism because of an unsuitable member arrangement or support configuration. Stability must therefore be established before the count is interpreted as a meaningful degree of static indeterminacy. Hence Option B is correct.
వివరణ:
Question 4
ప్రశ్న 4
For a stable plane pin-jointed truss having m members, j joints and r independent support-reaction components, which set of expressions is correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A plane truss has two independent joint-equilibrium equations at each joint. Its total static-indeterminacy count is therefore m + r - 2j when the truss is stable. Separating the three whole-structure equilibrium equations gives external indeterminacy r - 3. The remaining internal count is m - (2j - 3), and the two components sum to the total value. Therefore Option B is correct.
వివరణ:
Question 5
ప్రశ్న 5
A nondegenerate triangular plane truss has three pin-jointed members connecting three non-collinear joints. It is supported by a pin at one joint and a roller at another, giving three reaction components. What is its degree of static indeterminacy?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Here m = 3, j = 3 and r = 3. The plane-truss relation gives Ds = m + r - 2j = 3 + 3 - 6 = 0. The three non-collinear members form a geometrically stable triangle. The support arrangement also prevents rigid-body motion. Hence the truss is stable and statically determinate, so Option A is correct.
వివరణ:
Question 6
ప్రశ్న 6
Four pin joints form a nondegenerate rectangle and are connected only by its four side members. A pin and a roller provide three support reactions. What is the correct classification of this structure?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For this assemblage, m = 4, j = 4 and r = 3. The count m + r - 2j equals -1, indicating that the system has insufficient constraints rather than a meaningful negative degree of indeterminacy. More importantly, an unbraced pin-jointed rectangle can change shape into a parallelogram without changing member lengths. It therefore possesses an internal mechanism despite having adequate external supports. Hence Option C is correct.
వివరణ:
Question 7
ప్రశ్న 7
A nondegenerate rectangular plane truss has its four side members and one diagonal member. It is supported by a pin and a roller, and the diagonal properly triangulates the panel. What is its degree of static indeterminacy?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The truss has m = 5 members, j = 4 joints and r = 3 reaction components. Thus Ds = 5 + 3 - 2(4) = 0. The diagonal prevents the rectangular panel from deforming as a mechanism. The pin-and-roller support arrangement also provides external stability. Therefore the truss is stable and statically determinate, making Option A correct.
వివరణ:
Question 8
ప్రశ్న 8
A rectangular plane truss has four side members and both diagonals, with the diagonals crossing without forming an additional joint. It is supported by a pin and a roller. What are its external, internal and total static indeterminacies?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Here m = 6, j = 4 and r = 3. The external indeterminacy is r - 3 = 0. The internal indeterminacy is m - (2j - 3) = 6 - 5 = 1. Consequently the total indeterminacy is 1, which also follows from m + r - 2j. Therefore Option B is correct.
వివరణ:
Question 9
ప్రశ్న 9
The rectangular truss in the previous question is instead supported by two pins at distinct locations, giving four reaction components. Assume the geometry and support locations make the truss stable. What are the external, internal and total static indeterminacies?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The truss still has m = 6 members and j = 4 joints, but now r = 4. Its external indeterminacy is r - 3 = 1. Its internal indeterminacy remains 6 - (8 - 3) = 1 because changing the supports does not change the member redundancy. The total is therefore 1 + 1 = 2, also equal to 6 + 4 - 8. Hence Option C is correct.
వివరణ:
Question 10
ప్రశ్న 10
A stable triangulated plane truss has 6 joints, 9 members and 4 independent support-reaction components. What are its external, internal and total static indeterminacies?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The external indeterminacy is r - 3 = 4 - 3 = 1. The internal indeterminacy is m - (2j - 3) = 9 - 9 = 0. Hence the truss has no redundant member but has one redundant support-reaction component. The total indeterminacy is also 9 + 4 - 12 = 1. Therefore Option A is correct.
వివరణ:
Question 11
ప్రశ్న 11
A stable plane truss has 6 joints, 10 members and the usual pin-and-roller support arrangement giving 3 reaction components. What are its external, internal and total static indeterminacies?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
With r = 3, the external static indeterminacy is zero. For j = 6, an internally determinate stable truss would require 2j - 3 = 9 members. Since the actual truss has 10 members, its internal indeterminacy is one. The total count is 10 + 3 - 12 = 1. Therefore Option B is correct.
వివరణ:
Question 12
ప్రశ్న 12
When axial deformation of the members is included, what is the usual kinematic-indeterminacy count for a plane pin-jointed truss having j joints and r restrained translational displacement components at its supports?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Each joint of a plane truss has two possible translational displacement components. Joint rotation is not an independent truss degree of freedom because the members are connected by ideal pins and carry axial force. Before support restraints are imposed, there are therefore 2j translational coordinates. Subtracting the r restrained translational coordinates gives Dk = 2j - r. Hence Option B is correct.
వివరణ:
Question 13
ప్రశ్న 13
For the stable triangular plane truss with 3 joints, a pin support restraining two translations and a roller restraining one translation, what is its kinematic indeterminacy when member axial deformations are included?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A plane truss joint has two translational degrees of freedom. Three joints therefore provide six possible joint translations before supports are applied. The pin and roller together restrain three translational components. Thus Dk = 6 - 3 = 3. Therefore Option D is correct.
వివరణ:
Question 14
ప్రశ్న 14
A stable plane truss has 6 joints and supports that restrain 4 independent translational displacement components. If axial deformation of members is included, what is its kinematic indeterminacy?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Six plane-truss joints provide 2 × 6 = 12 translational displacement coordinates. The supports restrain four of these coordinates. The remaining number of independent joint displacement components is therefore 12 - 4 = 8. This kinematic count is conceptually different from the structure's static-indeterminacy count. Therefore Option C is correct.
వివరణ:
Question 15
ప్రశ్న 15
For a stable rigid-jointed plane frame with no internal hinges or member-end releases, having m members, j joints and r external reaction components, which expression gives the total degree of static indeterminacy?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A general plane-frame member can transmit three internal force resultants at an end: axial force, shear and bending moment. Counting member force unknowns and independent joint equilibrium equations leads to the standard expression Ds = 3m + r - 3j for a stable rigid-jointed plane frame without internal releases. The expression differs from the plane-truss relation because frame members transmit bending and shear as well as axial force. Special hinges or releases require corresponding modifications to the count. Therefore Option C is correct.
వివరణ:
Question 16
ప్రశ్న 16
A propped cantilever beam has a fixed support at A and a vertical roller support at B. Treating it as a stable plane structure, what is its degree of static indeterminacy?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The fixed support provides three reaction components in the plane. The roller provides one additional vertical reaction, so there are four external reaction components. A stable planar structure has three independent whole-structure equilibrium equations. One reaction is therefore redundant and must be determined using deformation compatibility. Hence the propped cantilever is statically indeterminate to degree one and Option B is correct.
వివరణ:
Question 17
ప్రశ్న 17
A beam is rigidly fixed at both ends and has no internal hinges. What is its degree of static indeterminacy when treated as a general plane structure?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Each fixed support supplies a horizontal reaction, a vertical reaction and a moment reaction. Thus the two fixed supports provide six reaction components. Only three independent equilibrium equations are available for the entire planar beam. The excess number of force unknowns is therefore 6 - 3 = 3. Hence a fixed-fixed beam is statically indeterminate to degree three and Option C is correct.
వివరణ:
Question 18
ప్రశ్న 18
A two-span continuous beam ABC has a pin support at A and vertical roller supports at B and C. There are no internal hinges. What is its degree of static indeterminacy?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The pin at A provides two reaction components. The rollers at B and C each provide one vertical reaction, giving four external reaction components in total. Three independent whole-structure equilibrium equations are available in the plane. Consequently one support reaction is redundant. The continuous beam is therefore statically indeterminate to degree one and Option B is correct.
వివరణ:
Question 19
ప్రశ్న 19
A one-bay rigid portal frame consists of two columns and one beam, giving m = 3 members and j = 4 joints including the two base joints. Both bases are fixed, so r = 6. There are no internal hinges or releases. What is its total static indeterminacy?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For a stable rigid-jointed plane frame without releases, Ds = 3m + r - 3j. Substitution gives Ds = 3(3) + 6 - 3(4). This equals 9 + 6 - 12 = 3. The three redundants are consistent with the restraint supplied by the two fixed bases. Therefore Option C is correct.
వివరణ:
Question 20
ప్రశ్న 20
A closed rectangular rigid-jointed plane frame ABCD has members AB, BC, CD and DA. It has a pin support at A and a vertical roller at D, so r = 3; there are no internal hinges or member releases. Assuming a stable nondegenerate geometry, what are its external, internal and total static indeterminacies?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The support arrangement has three reaction components, so there is no external support redundancy relative to the three planar equilibrium equations. Here m = 4, j = 4 and r = 3. The total frame indeterminacy is 3(4) + 3 - 3(4) = 3. Since the external indeterminacy is zero, these three redundants are internal and arise from closing the rigid frame loop. Therefore Option B is correct.
వివరణ:
Question 21
ప్రశ్న 21
Consider a one-bay fixed-base portal frame with two columns and one beam. If member axial deformations are neglected, which set correctly describes its usual kinematic indeterminacy for general in-plane bending and sidesway?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
With fixed bases, all displacement components at the two base joints are restrained. Neglecting axial deformation prevents independent axial shortening or extension of the columns and beam. The two upper joints can rotate independently, giving two rotational coordinates. They can also undergo a common horizontal sidesway of the portal. Therefore the usual kinematic indeterminacy is three and Option C is correct.
వివరణ:
Question 22
ప్రశ్న 22
For the same one-bay fixed-base portal frame, suppose axial deformation of all members is included. Treating each rigid frame joint as having two translations and one rotation, what is the kinematic indeterminacy?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A plane rigid-frame joint has three displacement coordinates when axial deformation is included: horizontal translation, vertical translation and rotation. The two fixed base joints have all three coordinates restrained. Each of the two upper joints therefore retains three independent displacement coordinates before structural equations are solved. This gives a total of 2 × 3 = 6 kinematic unknowns. Hence Option D is correct.
వివరణ:
Question 23
ప్రశ్న 23
For a propped cantilever beam fixed at A and vertically supported by a roller at B, what is its usual kinematic indeterminacy in elementary beam-bending analysis when axial deformation is neglected?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
At the fixed end A, both transverse displacement and rotation are restrained. At the roller end B, transverse displacement is also restrained. The rotation at B remains free and is the only unknown generalized displacement in the elementary flexural model. Axial displacement is excluded by the stated assumption that axial deformation is neglected. Therefore the kinematic indeterminacy is one and Option B is correct.
వివరణ:
Question 24
ప్రశ్న 24
A simply supported beam has a pin at A and a roller at B. In elementary beam-bending analysis with axial deformation neglected, what is its kinematic indeterminacy?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The transverse displacement at each simple support is zero. Neither the pin nor the roller restrains the beam-end rotation. Thus the end rotations at A and B are two independent flexural displacement coordinates. Axial translation is not counted because axial deformation is neglected in the stated beam model. Therefore the kinematic indeterminacy is two and Option C is correct.
వివరణ:
Question 25
ప్రశ్న 25
Consider the following statements:
1. Static indeterminacy counts independent force redundants.
2. Kinematic indeterminacy counts independent displacement coordinates.
3. A statically determinate structure may still have a positive kinematic indeterminacy.
4. If a numerical determinacy formula gives zero, the structure is necessarily stable regardless of its geometry.
Which statements are correct?
1. Static indeterminacy counts independent force redundants.
2. Kinematic indeterminacy counts independent displacement coordinates.
3. A statically determinate structure may still have a positive kinematic indeterminacy.
4. If a numerical determinacy formula gives zero, the structure is necessarily stable regardless of its geometry.
Which statements are correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Statement 1 is correct because static indeterminacy measures redundant forces or reactions. Statement 2 is correct because kinematic indeterminacy measures independent translations and rotations required in displacement analysis. Statement 3 is also correct, as a simply supported beam is statically determinate but has unknown end rotations in a flexural displacement formulation. Statement 4 is false because geometric instability or a mechanism can exist even when a counting equation produces zero. Therefore Statements 1, 2 and 3 are correct and Option B is the unique answer.
వివరణ:
Answer Key సమాధానాల పట్టిక
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Question 1 ప్రశ్న 1Answer: B. The number of independent force redundants that cannot be determined by equilibrium equations alone సమాధానం: B.
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Question 2 ప్రశ్న 2Answer: C. The number of independent joint displacement components required to describe the deformed configuration సమాధానం: C.
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Question 3 ప్రశ్న 3Answer: B. Because counting equations alone may give zero or a positive value even when the geometry permits a mechanism సమాధానం: B.
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Question 4 ప్రశ్న 4Answer: B. External = r - 3; internal = m - (2j - 3); total = m + r - 2j సమాధానం: B.
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Question 5 ప్రశ్న 5Answer: A. 0 సమాధానం: A.
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Question 6 ప్రశ్న 6Answer: C. Unstable because the rectangular assemblage can distort as a mechanism సమాధానం: C.
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Question 7 ప్రశ్న 7Answer: A. 0 సమాధానం: A.
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Question 8 ప్రశ్న 8Answer: B. External = 0, internal = 1, total = 1 సమాధానం: B.
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Question 9 ప్రశ్న 9Answer: C. External = 1, internal = 1, total = 2 సమాధానం: C.
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Question 10 ప్రశ్న 10Answer: A. External = 1, internal = 0, total = 1 సమాధానం: A.
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Question 11 ప్రశ్న 11Answer: B. External = 0, internal = 1, total = 1 సమాధానం: B.
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Question 12 ప్రశ్న 12Answer: B. 2j - r సమాధానం: B.
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Question 13 ప్రశ్న 13Answer: D. 3 సమాధానం: D.
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Question 14 ప్రశ్న 14Answer: C. 8 సమాధానం: C.
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Question 15 ప్రశ్న 15Answer: C. 3m + r - 3j సమాధానం: C.
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Question 16 ప్రశ్న 16Answer: B. 1 సమాధానం: B.
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Question 17 ప్రశ్న 17Answer: C. 3 సమాధానం: C.
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Question 18 ప్రశ్న 18Answer: B. 1 సమాధానం: B.
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Question 19 ప్రశ్న 19Answer: C. 3 సమాధానం: C.
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Question 20 ప్రశ్న 20Answer: B. External = 0, internal = 3, total = 3 సమాధానం: B.
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Question 21 ప్రశ్న 21Answer: C. Three degrees of freedom: two top-joint rotations and one common horizontal sidesway సమాధానం: C.
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Question 22 ప్రశ్న 22Answer: D. 6 సమాధానం: D.
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Question 23 ప్రశ్న 23Answer: B. 1 సమాధానం: B.
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Question 24 ప్రశ్న 24Answer: C. 2 సమాధానం: C.
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Question 25 ప్రశ్న 25Answer: B. 1, 2 and 3 only సమాధానం: B.