Govt Exams
For a frame to be stable and determinate: m + r = 2j. If m + r < 2j, the frame is unstable (insufficient members/supports). If m + r > 2j, frame is indeterminate.
A hinged/pin support prevents translation in both horizontal and vertical directions (2 restraints), thus has 2 unknown reactions (Hx and Vy). Rotation is free.
IS 800:2007 uses nonlinear interaction equations for combined bending and compression, typically: (P/Pc) + (M/Mc) ≤ 1.0, where Pc and Mc are critical values.
In method of sections for trusses, we can cut maximum 3 members (one truss has 3 equilibrium equations: ΣFx=0, ΣFy=0, ΣM=0) to find member forces.
Stiffness coefficient k_ij represents the force/moment at DOF i due to unit displacement at DOF j. k₁₂ means effect at location 1 from displacement at location 2.
Flexibility method removes redundant reactions/members equal to degree of indeterminacy to create a primary determinate structure, then applies compatibility equations.
For continuous beams with equal spans and uniform loading, the middle span typically experiences larger negative moments at supports and larger positive moments due to symmetry considerations.
Castigliano's second theorem: ∂U/∂P = deflection at point of load application. U is strain energy and P is the applied load.
Fixed-end moment for a fixed-fixed beam with central point load = ±WL/8. This is a standard formula used in slope-deflection equations.
DKI = 3n - r, where n = number of joints = 8, r = reaction components = 9 (3 fixed supports × 3). DKI = 3(8) - 9 = 24 - 9 = 15.