Govt Exams
∂U/∂P = Deflection at point of application of load P. ∂U/∂M = Slope at point of application of moment M.
Slope at free end of cantilever with UDL = wL³/(3EI). Deflection at free end = wL⁴/(8EI).
Maximum 3 members can be cut in a 2D truss to maintain a determinate section with 3 equilibrium equations available.
For a simply supported beam with central point load, maximum bending moment occurs at the center (L/2) and equals PL/4.
Influence line for reaction is linear: value = 1 at support A and linearly decreases to 0 at support B, representing unit load position effect on reaction RA.
Maximum bending moment occurs at the section where shear force is zero (point of contraflexure). For moving loads, position varies but this principle always applies.
Superposition principle applies only to linear elastic systems with small displacements where strain-displacement and stress-strain relationships are linear.
Plastic analysis assumes yielding at critical sections forms plastic hinges where full plastic moment capacity is achieved, leading to collapse mechanism.
For a parabolic arch under UDL, the horizontal thrust H = wL²/(8f), where f is rise. The thrust decreases with increase in rise-to-span ratio. Optimal ratio is 1/2 for many design considerations.
Conjugate beam method applies M/EI diagram as load on conjugate structure. Deflection and slope are obtained as reactions and moments on conjugate beam.