JNTU B.Tech III ADVANCED STRUCTURAL ANALYSIS Previous Question Paper (February/March - 2022 - R18)

Static Indeterminacy (D_s) refers to the number of unknown forces (reactions or internal forces) that cannot be determined using the equations of static equilibrium alone. It is categorized into external (D_se) and internal (D_si) indeterminacy.

For a plane frame, D_s = (3m + r) - 3j, where m is the number of members, r is the number of reaction components, and j is the number of joints. For example, a propped cantilever beam has 4 reaction components and 3 equilibrium equations, resulting in a static indeterminacy of 1. A fixed-ended beam has 6 reaction components, resulting in a static indeterminacy of 3.

Kinematic Indeterminacy (D_k), also known as the degree of freedom, refers to the number of independent joint displacements (translations and rotations) required to define the deformed shape of the structure.

For a plane frame, D_k = 3j - r_e, where r_e is the number of restrained coordinates. For example, a simply supported beam has 2 degrees of freedom (rotations at the supports), while a cantilever beam has 1 degree of freedom (rotation at the free end) if axial deformation is neglected.

Consider a truss element of length L, cross-sectional area A, and Young's modulus E. Let the axial displacements at the two ends be u₁ and u₂, and the corresponding axial forces be F₁ and F₂.

The axial strain is given by ε = (u₂ - u₁) / L. According to Hooke's Law, the axial stress is σ = Eε = E(u₂ - u₁) / L. The internal axial force is P = σA = (AE / L)(u₂ - u₁).

Applying equilibrium at the nodes:

In matrix form, the force-displacement relationship is:

This represents the stiffness matrix for a one-dimensional truss element.