In Castigliano's theorem, the partial derivative of total strain energy with respect to a load gives:
∂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:
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:
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:
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:
[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:
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:
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:
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:
Global stiffness matrix is assembled by superposition of element stiffness matrices after coordinate transformation.
In plastic hinge formation, the condition for plastic collapse is:
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:
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:
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:
Fixed-end moment for fixed beam with UDL = wL²/12 at each end (opposite signs).
The principle of virtual displacements is applicable to:
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:
DSI = (r + 3m) - 2j = (4 + 3×3) - 2×4 = 13 - 8 = 3 for this portal frame configuration.
What is the primary function of cement in concrete?
Cement acts as a binder that hydrates and binds fine and coarse aggregates together to form a solid mass.
According to IS 456:2000, what is the standard cube compressive strength test age for concrete?
IS 456:2000 specifies 28 days as the standard curing age for determining the compressive strength of concrete cubes.
Which type of cement is used for underwater construction?
SRPC (Sulphate Resisting Portland Cement) is used for underwater and marine structures to resist sulphate attack.
What is the water-cement ratio for M20 grade concrete as per IS 456:2000?
For M20 concrete, the maximum water-cement ratio is 0.55 to achieve the specified 28-day compressive strength.
Which of the following is NOT a type of aggregate used in concrete?
Rigid aggregate is not a standard classification. Aggregates are classified as coarse, fine, and lightweight based on size and density.
