In Castigliano's theorem, the partial derivative of total strain energy with respect to a load gives:
A Slope at that point B Deflection at that point C Bending moment at that point D Shear force at that point
∂U/∂P = Deflection at point of application of load P. ∂U/∂M = Slope at point of application of moment M.
A fixed beam of length L with central point load P has the fixed end moment equal to:
A PL/8 B PL/6 C PL/4 D PL/2
For fixed beam with central point load, fixed end moment = PL/8 and maximum deflection = PL³/(192EI).
In virtual work method, if a unit load is applied instead of actual load, the virtual displacement is equal to:
A Actual displacement B Zero displacement C Slope of the beam D Curvature of the beam
Virtual work principle: Work done by actual loads through virtual displacements equals work by virtual loads through actual displacements.
For a beam with varying moment of inertia, the flexibility coefficient method requires:
A Constant EI throughout B Integration with variable EI C Average value of EI D Maximum value of EI
Flexibility coefficients for non-uniform beams require integration accounting for variable EI along the length.
The stiffness matrix in matrix method of structural analysis represents:
A Load-displacement relationship B Displacement-force relationship C Moment-rotation relationship D Stress-strain relationship
[K]{D} = {F}, where stiffness matrix K relates displacements D to forces F. K represents member/structure stiffness.
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In a three-hinged arch, the additional hinge at crown provides:
A One additional equilibrium equation B Two additional constraints C Reduces indeterminacy to zero D Increases bending moment
Three-hinged arch has 4 reactions with 3 global + 1 local equilibrium equation at crown, making it statically determinate (DSI=0).
For a continuous beam with equal spans and equal loads, the negative moment at interior support is maximum at:
A First interior support B Second interior support C Middle interior support D All interior supports equally
For symmetrical continuous beam with equal spans and loads, negative moments at all interior supports are equal.
The radius of gyration for a rectangular section (b×d) about the centroidal axis parallel to base is:
A d/√12 B d/2√3 C b/√12 D √(bd/12)
I = bd³/12, A = bd, r = √(I/A) = √(d²/12) = d/(2√3) for axis parallel to base.
In the direct stiffness method, the global stiffness matrix is formed by:
A Assembly of element stiffness matrices B Averaging element stiffness matrices C Selection of largest element stiffness D Transformation of local to global coordinates only
Global stiffness matrix is assembled by superposition of element stiffness matrices after coordinate transformation.
In plastic hinge formation, the condition for plastic collapse is:
A Bending moment equals yield moment B Deflection becomes zero C Sufficient plastic hinges form to create mechanism D Shear force equals yield shear
Plastic collapse occurs when sufficient plastic hinges develop to convert the structure into a mechanism, allowing unbounded deformation.
For a beam element in finite element analysis, the shape functions are typically:
A Linear functions B Cubic Hermite polynomials C Quadratic functions D Trigonometric functions
Beam elements use cubic Hermite shape functions to ensure C¹ continuity (displacement and slope continuity).
The effective length factor K for a column fixed at both ends is:
A 2.0 B 1.0 C 0.5 D 0.7
For column fixed at both ends, effective length L_e = 0.5L (K=0.5), giving maximum buckling resistance.
In slope-deflection method, the fixed-end moment for a fixed beam with UDL w over length L is:
A wL²/12 B wL²/8 C wL²/6 D wL²/4
Fixed-end moment for fixed beam with UDL = wL²/12 at each end (opposite signs).
The principle of virtual displacements is applicable to:
A Determinate structures only B Indeterminate structures only C Both determinate and indeterminate structures D Rigid bodies only
Virtual work principle applies universally to all structures, both determinate and indeterminate, and rigid/deformable bodies.
For a portal frame with fixed bases and pinned roof, subjected to horizontal load, the degree of indeterminacy is:
A 2 B 3 C 4 D 6
DSI = (r + 3m) - 2j = (4 + 3×3) - 2×4 = 13 - 8 = 3 for this portal frame configuration.
In the method of sections, when analyzing a truss, the maximum number of members that can be cut in a single section is:
A 2 B 3 C 4 D 5
In the method of sections, we can cut maximum 3 members in a single section to ensure we have only 3 unknowns, which can be solved using 3 equilibrium equations (ΣFx=0, ΣFy=0, ΣM=0).
The deflection of a cantilever beam of length L carrying a point load W at the free end is:
A WL³/(3EI) B WL³/(4EI) C WL⁴/(3EI) D WL⁴/(8EI)
For a cantilever beam with point load at free end, maximum deflection = WL⁴/(3EI). This is a standard formula in structural analysis.
In Castigliano's theorem, the partial derivative of strain energy with respect to a force gives:
A Slope at that point B Deflection in the direction of that force C Moment at that point D Stress at that point
Castigliano's first theorem states that ∂U/∂P = δ, where U is strain energy and δ is deflection in the direction of force P.
The bending moment diagram for a simply supported beam carrying a uniformly distributed load is:
A Linear B Parabolic C Cubic D Hyperbolic
For UDL on simply supported beam, BMD is parabolic (second-degree curve) with maximum at center.
In a two-hinged arch, the number of unknown reactions is:
A 3 B 4 C 5 D 6
Two-hinged arch has 2 supports (each with 2 reactions) = 4 unknowns total (horizontal and vertical at each hinge).
