Test 40 - Matrix Methods of Structural Analysis

Question 1 ప్రశ్న 1

Which statement best describes the role of matrix methods in structural analysis?

A They replace equilibrium and compatibility by purely graphical procedures. B They organize equilibrium, compatibility and force-displacement relationships into systematic matrix equations suitable for structures with many degrees of freedom. C They are applicable only to statically determinate trusses. D They can be used only when every structural member has the same stiffness.

Question 2 ప్రశ్న 2

Which statement correctly distinguishes the flexibility method from the stiffness method of matrix structural analysis?

A The flexibility method normally takes redundant forces as primary unknowns, whereas the stiffness method normally takes independent joint displacements as primary unknowns. B Both methods always use exactly the same primary unknowns. C The flexibility method uses only joint rotations, whereas the stiffness method uses only support reactions. D The stiffness method is a force method, whereas the flexibility method is a displacement method.

Question 3 ప్రశ్న 3

For a structure having degree of static indeterminacy Ds and degree of kinematic indeterminacy Dk, which statement is generally correct for the basic matrix formulations?

A The flexibility matrix must always be of order Dk, while the reduced stiffness matrix must always be of order Ds. B Both matrices must always have order Ds + Dk. C The basic flexibility matrix is associated with Ds force unknowns, while the reduced structural stiffness matrix is associated with Dk displacement unknowns. D Neither matrix order is related to structural indeterminacy.

Question 4 ప్రశ్న 4

In a one-redundant flexibility-method problem, the released primary structure undergoes a displacement of +3 mm at the redundant coordinate due to the applied loads. The flexibility coefficient at that coordinate is 0.5 mm/kN. If the original support condition requires zero displacement there, what redundant force X is required?

A +1.5 kN B +6 kN C -6 kN D -1.5 kN

Question 5 ప్రశ్న 5

Which matrix equation represents the fundamental equilibrium relation of the linear stiffness method after the structural degrees of freedom have been appropriately defined?

A [K]{d} = {P} B [K]{P} = {d} C [F]{d} = {P} D [K] + {d} = {P}

Question 6 ప్రశ్న 6

In a flexibility matrix [F], what does a coefficient fij represent under the usual generalized-coordinate definition?

A The force at coordinate i caused by a unit displacement at coordinate j B The displacement at coordinate i caused by a unit force applied at coordinate j C The stiffness of member i divided by the stiffness of member j D The reaction at support i caused by removing support j

Question 7 ప్రశ్న 7

For a stable linear-elastic structure with compatible reciprocal coordinates, which statement about its stiffness and flexibility matrices is correct?

A Both are generally symmetric, and when defined over the same nonsingular coordinate set, [F] = [K]^-1. B The stiffness matrix must be antisymmetric. C The flexibility matrix can never be inverted. D The diagonal terms of the stiffness matrix must all be zero.

Question 8 ప్రశ్న 8

For a two-node prismatic axial bar of length L, area A and Young's modulus E, which local stiffness matrix relates the axial nodal forces to the two local axial nodal displacements?

A (EI/L) [[1, 1], [1, 1]] B (AE/L²) [[1, -1], [-1, 1]] C (AE/L) [[1, -1], [-1, 1]] D (AE/L) [[1, 1], [-1, -1]]

Question 9 ప్రశ్న 9

A steel truss member has E = 200 GPa, A = 600 mm² and L = 3000 mm. What is its axial stiffness parameter AE/L?

A 20 kN/mm B 40 kN/mm C 60 kN/mm D 120 kN/mm

Question 10 ప్రశ్న 10

The truss member in the previous question is oriented at 45 degrees to the global x-axis. For its standard 4 × 4 global stiffness matrix, what is the coefficient relating global x-force at its first node to global x-displacement at the same node, considering this element alone?

A 10 kN/mm B 20 kN/mm C 28.28 kN/mm D 40 kN/mm

Question 11 ప్రశ్న 11

Let a transformation matrix [T] relate an element's global nodal displacement vector to its local displacement vector by {d'} = [T]{d}. Which expression gives the corresponding element stiffness matrix in global coordinates?

A [k] = [T]^T [k'] [T] B [k] = [T] [k'] [T]^T for every transformation definition C [k] = [k'] + [T] D [k] = [T]^-1 + [k']

Question 12 ప్రశ్న 12

Three collinear nodes 1, 2 and 3 are connected by two axial elements. Element 1-2 has stiffness k1 = 100 kN/mm and element 2-3 has stiffness k2 = 200 kN/mm. Which assembled global stiffness matrix is correct for nodal displacement ordering {u1, u2, u3}?

A [[100, -100, 0], [-100, 300, -200], [0, -200, 200]] kN/mm B [[100, 100, 0], [100, 300, 200], [0, 200, 200]] kN/mm C [[300, -100, -200], [-100, 100, 0], [-200, 0, 200]] kN/mm D [[100, 0, -100], [0, 200, -200], [-100, -200, 300]] kN/mm

Question 13 ప్రశ్న 13

For the three-node axial system in the previous question, nodes 1 and 3 are fixed so that u1 = u3 = 0. A horizontal load of +30 kN acts at node 2. What is u2?

A 0.05 mm B 0.10 mm C 0.15 mm D 0.30 mm

Question 14 ప్రశ్న 14

For the same three-node axial system, take forces to the right as positive. With u2 = +0.10 mm and no externally applied loads at the restrained nodes, what are the support reactions at nodes 1 and 3?

A R1 = +10 kN and R3 = +20 kN B R1 = -10 kN and R3 = -20 kN C R1 = -20 kN and R3 = -10 kN D R1 = +30 kN and R3 = 0

Question 15 ప్రశ్న 15

A reduced structural stiffness equation is [[30, -10], [-10, 20]]{d1, d2} = {20, 10}, where stiffnesses are in kN/mm and loads are in kN. What are d1 and d2?

A d1 = 0.5 mm and d2 = 1.0 mm B d1 = 1.0 mm and d2 = 0.5 mm C d1 = 1.0 mm and d2 = 1.0 mm D d1 = 2.0 mm and d2 = 1.0 mm

Question 16 ప్రశ్న 16

Why can the fully assembled stiffness matrix of a free, unsupported structural assemblage be singular before support boundary conditions are imposed?

A Because rigid-body displacement modes can occur without developing restoring forces. B Because every element stiffness matrix must have negative diagonal terms. C Because matrix assembly eliminates equilibrium. D Because a supported structure must always have a zero determinant even after restraints are applied.

Question 17 ప్రశ్న 17

How are prescribed support displacements such as zero translations at fixed or pinned supports normally handled in the direct stiffness method?

A They are ignored until after all member forces are calculated. B They are imposed as boundary conditions, and the equilibrium equations are reduced or partitioned to solve for the unknown free displacements. C They are treated as unknown redundant forces. D Every restrained displacement is automatically replaced by a unit displacement.

Question 18 ప్రశ్న 18

For an Euler-Bernoulli prismatic beam element having local bending degrees of freedom {vi, θi, vj, θj}, what is the diagonal translational stiffness coefficient associated with vi in the standard local bending stiffness matrix?

A EI/L B 4EI/L C 6EI/L² D 12EI/L³

Question 19 ప్రశ్న 19

A prismatic beam element has EI = 24000 kN·m² and L = 4 m. What is the diagonal rotational stiffness coefficient 4EI/L at either end in its local bending stiffness matrix?

A 6000 kN·m/rad B 12000 kN·m/rad C 24000 kN·m/rad D 96000 kN·m/rad

Question 20 ప్రశ్న 20

In a two-dimensional rigid-jointed frame formulation including axial and flexural deformation, how many nodal degrees of freedom are normally assigned to each joint, and what is the order of the local stiffness matrix of a two-node frame element?

A Two degrees of freedom per joint and a 4 × 4 element matrix B Three degrees of freedom per joint and a 6 × 6 element matrix C Four degrees of freedom per joint and an 8 × 8 element matrix D Six degrees of freedom per joint and a 12 × 12 element matrix

Question 21 ప్రశ్న 21

In a standard plane pin-jointed truss formulation, how many displacement degrees of freedom are normally assigned to each joint, and what is the size of the global-coordinate stiffness matrix of one two-node truss member?

A One degree of freedom per joint and a 2 × 2 global matrix B Two translational degrees of freedom per joint and a 4 × 4 global element matrix C Three degrees of freedom per joint and a 6 × 6 global matrix D Two rotational degrees of freedom per joint and a 4 × 4 global matrix

Question 22 ప్రశ్న 22

A plane truss member has direction cosines c = 0.6 and s = 0.8. The displacement of its second node relative to its first is Δu = +2 mm in the global x-direction and Δv = +1 mm in the global y-direction. What is the axial extension of the member?

A 0.4 mm B 1.4 mm C 2.0 mm D 2.8 mm

Question 23 ప్రశ్న 23

A two-coordinate flexibility matrix is [F] = [[0.3, 0.1], [0.1, 0.2]] mm/kN. What is the corresponding stiffness matrix [K] = [F]^-1?

A [[3, -1], [-1, 2]] kN/mm B [[4, -2], [-2, 6]] kN/mm C [[6, 2], [2, 4]] kN/mm D [[0.3, -0.1], [-0.1, 0.2]] kN/mm

Question 24 ప్రశ్న 24

Two axial elements meet at node 2. Their axial stiffness parameters are k1 = 100 kN/mm and k2 = 200 kN/mm, and each contributes its near-node stiffness to the same global displacement coordinate u2. What is the assembled diagonal stiffness contribution K22 from these two elements?

A 100 kN/mm B 200 kN/mm C 300 kN/mm D 20000 kN/mm

Question 25 ప్రశ్న 25

Consider the following statements about the direct stiffness method:
1. Element stiffness contributions associated with a common structural degree of freedom are added during assembly.
2. Prescribed support displacements are imposed as boundary conditions before the unknown free displacements are finally solved.
3. Member matrices for differently oriented members must be expressed in compatible global coordinates before they can be assembled into one structural matrix.
4. The flexibility method normally uses joint displacements rather than redundant forces as its basic unknowns.

Which statements are correct?

A 1 and 4 only B 1, 2 and 3 only C 2, 3 and 4 only D 1, 2, 3 and 4

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