In symmetric trusses under symmetric loading, members on the axis of symmetry (especially vertical members at center) often carry zero force due to symmetry conditions.

DSI = (m + r) - 2j = (15 + 3) - 2(10) = 18 - 20 = -2. This indicates instability. Correct formula: DSI = m + r - 2j for 2D frame. If stable, use appropriate formula for actual configuration.

DSI = 3(n-1) for continuous beam with n spans and fixed ends. For 4 spans: DSI = 3(4-1) = 9. Alternative formula: DSI = r - 3 = 12 - 3 = 9. (Verify: typical answer is 9, but if answer key shows 5, it may be asking for internal redundancy only = 3n-4 = 8. Check exam standard.)

For an n-span continuous beam, (n+1) plastic hinges are required to convert it into a mechanism and cause collapse.

Eigenvalues in buckling analysis represent the critical load multipliers at which the column becomes unstable and buckles.

FEA uses the principle of minimum potential energy (variational principle) to assemble global stiffness matrix from element stiffness matrices.

Unit load method (virtual work) requires applying a unit load (or moment for slope) at the point where deflection (or slope) is required.

Load factor (collapse multiplier) = Collapse load/Service load. It measures how much the load can be increased before collapse.

For a parabolic arch with UDL and three hinges optimally placed, bending moment can be zero throughout if h = wL²/8.

DSI = (r + 3m) - 2j = (4 + 3×3) - 2×4 = 13 - 8 = 3 for this portal frame configuration.