Structural Analysis
Structural analysis finds internal forces and deflections in beams, frames and trusses. Learn determinate vs indeterminate structures, continuous beams, fixed-end moments, moment distribution, slope-deflection, the stiffness (matrix) method and influence lines.
Most real structures — continuous beams, rigid frames, multi-storey buildings — have more supports and connections than equilibrium alone can resolve. They are statically indeterminate, and their internal forces depend on stiffness as well as loads.
Structural analysis combines equilibrium, compatibility (how parts deform together) and material behaviour to find bending moments, shears, axial forces and deflections everywhere — the input to every design check.
An analogy
Several people carrying a long plank share the load according to how firmly each holds it: a stiffer support attracts more load. Indeterminate structures share load by stiffness.
Why indeterminate?
Extra supports and rigid joints make structures stiffer, reduce peak moments and give redundancy — if one member is damaged, loads find another path. The price is that analysis needs compatibility equations or a computer.
Classical methods
Force (flexibility) method: remove redundants, compute deflections, restore compatibility. Slope-deflection: express member end moments in terms of joint rotations. Moment distribution (Hardy Cross): lock all joints, apply fixed-end moments, then release and distribute unbalanced moments by stiffness, carrying half over to far ends — iterate until balanced.
Influence lines
For moving loads (vehicles, cranes) an influence line shows how a reaction, shear or moment at one point varies as a unit load travels along the structure. Multiplying by axle loads (or integrating a UDL) gives the maximum effect and the critical load position.
Matrix stiffness method and software
Each member's stiffness matrix is assembled into [K]; solving [K]{d} = {F} gives joint displacements, then member forces. This is what STAAD.Pro, ETABS, SAP2000 and Robot do — plus P-Δ (second-order), nonlinear, dynamic and staged-construction analyses. Engineers must still check results with hand methods.
Formulas
- Fixed-end moments (UDL):
FEM = ± wL²/12 - Fixed-end moments (central point load):
FEM = ± PL/8 - Two equal spans, UDL:
M_support = −wL²/8, R = 3wL/8, 10wL/8, 3wL/8 - Distribution factor:
DF_i = k_i / Σk, carry-over = ½ - Indeterminacy (beams):
D_s = r − 3 (r = reactions)
Worked example: Two-span continuous beam
Given: Two equal spans L = 6 m; UDL w = 20 kN/m
Find: Reactions and moments.
- Support moment = −wL²/8 = −90 kN·m
- End reactions = 3wL/8 = 45 kN; middle = 10wL/8 = 150 kN
- Max sagging at 3L/8 = 2.25 m: 45²/(2 × 20) = 50.6 kN·m
Answer: R = 45/150/45 kN; M = −90 (support), +50.6 kN·m (span)
Worked example: Fixed-end moment
Given: Fixed-fixed beam L = 6 m, w = 20 kN/m
Find: End and mid-span moments.
- FEM = wL²/12 = 60 kN·m (hogging)
- Mid-span = wL²/24 = 30 kN·m
Answer: −60 at ends, +30 kN·m at mid-span
Common mistakes
- Wrong: Designing continuous beams as simply supported. — Right: Provide hogging (top) reinforcement over supports.
- Wrong: Trusting software output blindly. — Right: Check equilibrium (sum of reactions = loads) and approximate moments by hand.
Applications
- Continuous bridges & floors: Continuity reduces span moments and deflections.
- Rigid frames: Portal frames in industrial sheds, multistorey moment frames.
- Software checks: Hand methods verify FE model results for sanity.
Key takeaways
- Indeterminate structures share load by stiffness.
- Know FEMs: wL²/12, PL/8.
- Moment distribution: distribute by stiffness, carry over ½.
- Matrix stiffness method powers all analysis software.
Going further
- Second-order (P-Δ) analysis: Axial loads acting on deflected shapes amplify moments in slender frames.
- Plastic analysis: Collapse mechanisms with plastic hinges give ultimate capacity of ductile steel frames.
- Finite element analysis: Plates and shells (slabs, walls, cores) are modelled with 2D elements for realistic load sharing.