structural-analysis

Structural Analysis

Structural analysis is the determination of forces, stresses, and deformations in engineered structures under load. It is the foundation of structural engineering and a critical input to every design decision involving physical artifacts that must bear weight, resist wind, survive earthquakes, or simply not fall down. This skill covers the analytical toolkit from free-body diagrams through finite element concepts, with worked examples and failure case studies.

Agent affinity: roebling (structural and civil engineering), watt (mechanical and thermal loading)

Concept IDs: engr-stress-strain, engr-material-selection, engr-structural-systems, engr-structural-failure

The Structural Analysis Toolbox

| # | Topic | Key question | |---|---|---| | 1 | Statics and equilibrium | Is the structure in balance? | | 2 | Internal forces | What forces act inside the structure? | | 3 | Stress and strain | How does the material respond to internal forces? | | 4 | Material properties | What can the material withstand? | | 5 | Beam analysis | How do beams carry loads? | | 6 | Column buckling | When do slender members fail by instability? | | 7 | Failure modes | How and why do structures fail? | | 8 | Factor of safety | How much margin is enough? | | 9 | FEA concepts | How do computers solve structural problems? |

Topic 1 -- Statics and Equilibrium

A structure is in static equilibrium when all forces and moments sum to zero. This is the starting point of every structural analysis.

Equilibrium equations (2D):

• Sum of forces in x = 0
• Sum of forces in y = 0
• Sum of moments about any point = 0

Three equations, three unknowns. A structure with more than three unknown reaction forces requires additional equations (compatibility, material behavior) and is called statically indeterminate.

Free-body diagram (FBD). Isolate the structure from its supports. Replace supports with reaction forces. Show all applied loads. The FBD is the single most important tool in structural analysis. Every error in a structural calculation can be traced to an incomplete or incorrect FBD.

Worked example. A simply supported beam of length L carries a point load P at midspan. Find the reactions.

Draw the FBD: Left support provides vertical reaction R_A. Right support provides vertical reaction R_B. The beam has no horizontal loads, so horizontal reactions are zero.

Sum of vertical forces: R_A + R_B - P = 0. Sum of moments about A: R_B L - P (L/2) = 0, so R_B = P/2. Therefore R_A = P/2.

By symmetry this is obvious, but the method works for any loading.

Topic 2 -- Internal Forces

Once external reactions are known, cut the structure at any section to find internal forces: axial force (N), shear force (V), and bending moment (M).

Sign convention. Positive shear: the portion to the left of the cut moves up relative to the right. Positive moment: the beam bends concave-up (sagging). Consistency matters more than convention -- pick one and hold it.

Shear and moment diagrams. Plot V and M along the length of the beam. These diagrams reveal where the structure experiences maximum internal forces and guide design decisions about cross-section sizing and reinforcement placement.

Key relationships:

• dV/dx = -w(x), where w is the distributed load intensity
• dM/dx = V
• The moment is maximum where shear passes through zero

Worked example. Uniformly distributed load w on a simply supported beam of length L.

Reactions: R_A = R_B = wL/2 (by symmetry). Shear at x from left support: V(x) = wL/2 - wx. V = 0 at x = L/2. Moment at x: M(x) = (wL/2)x - (w/2)x^2. Maximum moment at x = L/2: M_max = wL^2/8.

This formula -- wL^2/8 -- is one of the most commonly used results in structural engineering.

Topic 3 -- Stress and Strain

Stress is internal force per unit area. Strain is deformation per unit length.

| Type | Definition | Units | |---|---|---| | Normal stress (sigma) | Force perpendicular to cross-section / area | Pa (N/m^2) or psi | | Shear stress (tau) | Force parallel to cross-section / area | Pa or psi | | Normal strain (epsilon) | Change in length / original length | dimensionless | | Shear strain (gamma) | Angular distortion | radians |

Hooke's Law. For linear elastic materials: sigma = E * epsilon, where E is the elastic modulus (Young's modulus). This is the constitutive equation that connects stress (equilibrium) to strain (deformation).

Bending stress. In a beam: sigma = -M*y/I, where M is the bending moment, y is the distance from the neutral axis, and I is the second moment of area (moment of inertia). Maximum stress occurs at the extreme fiber (top or bottom of the beam).

Shear stress in beams. tau = V*Q/(I*b), where Q is the first moment of area above (or below) the point of interest, and b is the width at that point. Maximum shear stress occurs at the neutral axis.

Topic 4 -- Material Properties

Every material has properties that determine its structural behavior.

| Property | Symbol | What it measures | |---|---|---| | Elastic modulus | E | Stiffness (resistance to elastic deformation) | | Yield strength | sigma_y | Onset of permanent deformation | | Ultimate tensile strength | sigma_u | Maximum stress before fracture | | Poisson's ratio | nu | Lateral contraction under axial load | | Density | rho | Mass per unit volume | | Thermal expansion coefficient | alpha | Length change per degree temperature change | | Fracture toughness | K_Ic | Resistance to crack propagation |

Material Selection

Material selection matches material properties to design requirements. The Ashby chart (property vs. property plot) is the standard tool. Common trade-offs:

• Strength vs. weight: Aerospace demands high specific strength (sigma_y / rho). Aluminum alloys, titanium, carbon fiber composites.
• Strength vs. cost: Civil infrastructure favors cost-effective strength. Structural steel, reinforced concrete.
• Stiffness vs. weight: Precision instruments need high specific stiffness (E / rho).
• Ductility vs. strength: Ductile materials (mild steel) give warning before failure. Brittle materials (cast iron, glass) do not.

The ductility imperative. In structural engineering, ductile failure modes are preferred over brittle ones because ductile failure provides visible warning (deformation) before collapse. Design codes enforce this by requiring ductile materials and ductile connection details.

Topic 5 -- Beam Analysis

Beams are the workhorses of structural engineering. Common boundary conditions:

| Type | Reactions | Determinacy | |---|---|---| | Simply supported | 2 vertical, 1 horizontal | Determinate | | Cantilever | 1 vertical, 1 horizontal, 1 moment | Determinate | | Fixed-fixed | 2 vertical, 2 horizontal, 2 moments | Indeterminate | | Continuous (multi-span) | Multiple vertical reactions | Indeterminate |

Deflection. Beam deflection is found by double integration of the moment-curvature relationship: EI * d^2y/dx^2 = M(x). Standard results:

| Loading | Maximum deflection | |---|---| | Point load P at midspan, simply supported | PL^3 / (48EI) | | Uniform load w, simply supported | 5wL^4 / (384EI) | | Point load P at free end, cantilever | PL^3 / (3EI) |

Deflection limits. Codes limit deflection to prevent serviceability problems (cracking of finishes, discomfort, damage to non-structural elements). Common limit: L/360 for floor beams under live load.

Topic 6 -- Column Buckling

Slender compression members can fail by buckling -- a sudden lateral deflection -- at loads well below the material's compressive strength.

Euler's critical load. P_cr = pi^2 * EI / (KL)^2, where K is the effective length factor (depends on end conditions: K=1 for pinned-pinned, K=0.5 for fixed-fixed, K=2 for fixed-free, K=0.7 for fixed-pinned) and L is the column length.

Slenderness ratio. KL/r, where r is the radius of gyration (r = sqrt(I/A)). High slendernes