JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III ADVANCED STRUCTURAL ANALYSIS Previous Question Paper (August - 2022 - R18)

Static Indeterminacy (Ds) is the number of unknown reactions or internal forces that cannot be determined using the equations of static equilibrium alone. It is defined as Ds = Total unknowns - Equilibrium equations. For a propped cantilever beam with one fixed end and one roller support, there are four reaction components and three equilibrium equations, resulting in Ds = 4 - 3 = 1.

Kinematic Indeterminacy (Dk), also known as the degree of freedom, is the number of independent joint displacements (translations and rotations) required to define the deformed shape of a structure. For a fixed-ended beam, Dk is zero as there is no displacement or rotation at the supports. For a cantilever beam, the free end has three degrees of freedom (horizontal, vertical, and rotation), so Dk = 3.

Consider a beam element of length L, Young's modulus E, and moment of inertia I. The force-displacement relationship is derived using the stiffness matrix method, relating nodal forces {F} to nodal displacements {d}.

The nodal forces at each end consist of shear forces and bending moments. For a beam element with two nodes, each having a vertical displacement (v) and a rotation (θ), the stiffness matrix [k] is a 4x4 matrix.

The relationship is expressed as:

Where {F} = [V1, M1, V2, M2]ᵀ and {d} = [v1, θ1, v2, θ2]ᵀ. This matrix is derived by applying unit displacements at each degree of freedom and calculating the resulting forces using the slope-deflection method or the principle of virtual work.