Civil Engineering

For fixed-fixed beam with central point load, max deflection δ_max = WL³/(192EI) occurs at center. (Note: Verify exact formula for your reference.)

Castigliano's theorem: Deflection δ_i = ∂U/∂P_i, where U is strain energy and P_i is load at point i.

Under uniformly distributed horizontal load (common assumption in engineering), cable takes parabolic shape. Catenary is the actual shape under its own weight.

In 2D beam/frame element, each node has 3 DOF: horizontal displacement (u), vertical displacement (v), and rotation (θ). Total DOF per element = 6.

R_A = 50×8/12 = 33.33 kN; R_B = 50×4/12 = 16.67 kN. Just right of load: SF = 33.33 - 50 = -16.67 kN (or 16.67 kN downward).

Virtual work principle: For equilibrium, ΣδW_external + ΣδW_internal = 0. External work by real forces through virtual displacements equals internal work.

For a 2D truss: m = number of members, n = number of joints. Condition m = 2n - 3 ensures static determinacy (3 reactions for 2D frame).

Influence line for shear force in a segment without concentrated loads remains constant (horizontal). Between segments with point loads, it's linear.

Portal frame columns experience both axial forces (from vertical loads) and bending moments (from lateral loads and frame action), resulting in combined stresses.

Method of sections allows cutting maximum 3 members in a 2D truss because we have 3 equilibrium equations (ΣFx=0, ΣFy=0, ΣM=0) to solve for 3 unknowns.