Structural Analysis Fundamentals
Coverage matrix
What this course covers, section by section — including what it covers only briefly and what it leaves out.
The syllabus was built by auditing an established structural analysis curriculum of 18 chapters and listing every subsection, so that the gaps would be visible rather than invisible. 126 subsections are tracked here. 124 of them are covered by a written lesson.
A row is marked covered in full only when it has an explanation, a worked example or practice, and the numbers are recomputed by the test suite from the engineering library. Covered briefly means the idea is explained but without its own worked treatment. Not covered means exactly that — the topic is in the field but not in these lessons.
The reference curriculum is used privately, to decide scope. No text, diagram, example or problem from it is reproduced: every explanation, derivation, figure and question here is original.
- Subsections tracked
- 126
- Covered in full
- 122
- Covered briefly
- 2
- Not covered
- 0
A further 2 subsections are deliberately excluded, with the reason recorded against each one.
01Introduction
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 1.1 | Function of a structure | Covered in full | Module 1 · what a structure does |
| 1.2 | Structural forms | Covered in fullinteractive | Module 1 · what a structure does |
| 1.3 | Support systems | Covered in fullinteractive | Module 1 · from structure to model |
| 1.4 | Statically determinate and indeterminate structures | Covered in fullinteractive | Module 1 · from structure to model |
| 1.5 | Analysis and designCovered as context; design practice is deliberately out of scope. | Covered briefly | Module 1 · from structure to model |
| 1.6 | Structural idealisation | Covered in fullinteractive | Module 1 · from structure to model |
02Principles of Statics
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 2.1 | Force | Covered in fullderivation · 1 worked example · interactive | Module 2 · forces and resolving |
| 2.2 | Moment of a force | Covered in fullderivation · 1 worked example | Module 2 · moments and equilibrium |
| 2.3 | Resultant of a system of parallel forcesResultant of parallel forces: magnitude by summation, position by moments, checked from two different centres. | Covered in full1 worked example | Module 2 · moments and equilibrium |
| 2.4 | Equilibrium of force systems | Covered in fullderivation · 1 worked example · interactive | Module 2 · moments and equilibrium |
| 2.5 | Calculation of support reactionsVerified by solveBeam + equilibriumCheck. | Covered in full1 worked example · interactive | Module 3 · finding reactions |
03Normal Force, Shear Force, Bending Moment and Torsion
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 3.1 | Types of load | Covered in full | Module 3 · supports and free bodies |
| 3.2 | Notation and sign convention | Covered in full1 worked example | Module 4 · sign conventions |
| 3.3 | Normal forceNormal force from the cut-section method, and the stepped normal force diagram of a multi-storey column. | Covered in full1 worked example | Module 3 · the cut section method |
| 3.4 | Shear force and bending moment | Covered in fullderivation · 1 worked example · interactive | Module 4 · the cut section method |
| 3.5 | Load, shear force and bending moment relationshipsdV/dx = -w and dM/dx = V derived; verified against solveBeam. | Covered in fullderivation · 1 worked example · interactive | Module 3 · load shear moment relationships |
| 3.6 | TorsionTorque diagrams built as a series of steps, with the closure check and the design torque. | Covered in full1 worked example | Module 11 · torque diagrams and plastic torsion |
| 3.7 | Principle of superpositionSuperposition stated with both of its conditions, and applied to combine axial and bending stresses. | Covered in full1 worked example | Module 3 · the cut section method |
04Analysis of Pin-jointed Trusses
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 4.1 | Types of truss | Covered in fullinteractive | Module 4 · truss assumptions and determinacy |
| 4.2 | Assumptions in truss analysis | Covered in fullderivation | Module 4 · truss assumptions and determinacy |
| 4.3 | Idealisation of a truss | Covered in fullinteractive | Module 4 · truss assumptions and determinacy |
| 4.4 | Statical determinacym + r = 2j taught as necessary but not sufficient. | Covered in fullderivation | Module 4 · truss assumptions and determinacy |
| 4.5 | Resistance of a truss to shear force and bending momentChord force = M/d and diagonal force = V/sin(theta), derived from the equivalent beam. | Covered in full1 worked example | Module 4 · trusses beyond the basics |
| 4.6 | Method of jointsVerified by solveTruss (Gaussian elimination). | Covered in fullderivation · 1 worked example · interactive | Module 4 · joints and sections |
| 4.7 | Method of sectionsMethod of sections with a dedicated worked example, cross-checked against the beam analogy and against solveBeam. | Covered in full1 worked example | Module 4 · joints and sections |
| 4.8 | Method of tension coefficientsTension coefficients t = T/L, and why they extend to three dimensions unchanged. | Covered in full1 worked example | Module 4 · trusses beyond the basics |
| 4.9 | Graphical method of solutionHistorical method; superseded for teaching purposes by the interactive joint solver. Recorded as a deliberate exclusion. | Deliberately excluded | Module 4 |
| 4.10 | Compound trussesCompound trusses: determinacy count, why the method of joints stalls, and the section that restarts it. | Covered in full1 worked example | Module 4 · trusses beyond the basics |
| 4.11 | Pin-jointed space framesm + r = 3j and a symmetric tripod. Verified against tripodLegForce and spaceFrameDeterminacy. | Covered in full1 worked example | Module 4 · trusses beyond the basics |
05Cables
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 5.1 | Lightweight cables carrying concentrated loadsHorizontal component constant derived from a cable element. Verified against cableWithPointLoads. | Covered in fullderivation · 1 worked example | Module 5 · cables with point loads |
| 5.2 | Heavy cablesParabolic profile derived; the catenary is contrasted with it and the error of the parabolic approximation quantified. Verified against parabolicCable and catenaryCable. | Covered in fullderivation · 1 worked example | Module 5 · parabolic and catenary cables |
06Arches
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 6.1 | The linear archFunicular shape and the inverted-cable analogy, with the abutment thrust made explicit. | Covered in full | Module 6 · the cable arch analogy |
| 6.2 | The three-pinned archCrown hinge supplies the fourth equation. Verified against threePinnedArch. | Covered in fullderivation · 2 worked examples | Module 6 · three pinned arches |
| 6.3 | Three-pinned parabolic arch under uniform loadShown to carry a uniform load with zero bending everywhere. Verified against parabolicArchY and threePinnedArch. | Covered in fullderivation · 1 worked example | Module 6 · three pinned arches |
| 6.4 | Bending moment diagram for a three-pinned archBending moment plotted along the arch axis, with the crown-hinge zero and the sign reversal. Verified against solveBeam. | Covered in full1 worked example | Module 6 · three pinned arches |
07Stress and Strain
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 7.1 | Direct stress in tension and compression | Covered in fullderivation · 1 worked example · interactive | Module 7 · stress and strain |
| 7.2 | Shear stress in shear and torsionShear stress defined against direct stress, with the area distinction and a bolt in single and double shear. | Covered in full1 worked example · interactive | Module 7 · shear strain and poisson |
| 7.3 | Complementary shear stressComplementary shear derived as the reason a vertical shear force implies a horizontal shear stress. | Covered in full | Module 10 · why shear stress exists |
| 7.4 | Direct strain | Covered in fullderivation · 1 worked example · interactive | Module 7 · stress and strain |
| 7.5 | Shear strainShear stress vs direct stress, shear strain as an angle change, and tau = G.gamma. Verified against sectionProps and directStress. | Covered in full1 worked example · interactive | Module 7 · shear strain and poisson |
| 7.6 | Volumetric strain due to hydrostatic pressureBulk modulus and volumetric strain, including why nu cannot exceed 0.5. Verified against bulkModulus. | Covered in full1 worked example | Module 8 · elastic constants |
| 7.7 | Stress–strain relationships | Covered in fullderivation · 1 worked example · interactive | Module 7 · hookes law and stiffness |
| 7.8 | The Poisson effectPoisson's ratio, lateral strain, and why lateral stress only arises under restraint. | Covered in full1 worked example | Module 7 · shear strain and poisson |
| 7.9 | Relationships between the elastic constantsG = E/2(1+nu) and K = E/3(1-2nu), used as a consistency check on material data. Verified against shearModulus, bulkModulus and poissonFromEG. | Covered in full1 worked example | Module 8 · elastic constants |
| 7.10 | Strain energy in tension or compressionU = P2.L/2AE derived from the area under the load-deflection line. Verified against axialStrainEnergy. | Covered in fullderivation · 1 worked example | Module 15 · work and strain energy |
| 7.11 | Impact loads on structural membersImpact factor n = 1 + sqrt(1 + 2h/delta_st), including the factor of 2 for a suddenly applied load. Verified against impactFactor. | Covered in full | Module 8 · temperature restraint and impact |
| 7.12 | Deflections of axially loaded members | Covered in fullderivation · 1 worked example · interactive | Module 7 · hookes law and stiffness |
| 7.13 | Deflection of a simple trussUnit-load method, delta = sum(NnL/AE), applied to a two-member bracket. Verified against unitLoadTrussDeflection. | Covered in fullderivation · 1 worked example | Module 15 · virtual work and the unit load |
| 7.14 | Statically indeterminate axial systemsParallel members sharing load by compatibility. Verified against parallelAxialSharing; the stress ratio is checked against the modular ratio. | Covered in full1 worked example | Module 7 · pressure vessels and load sharing |
| 7.15 | Thin-walled shells under internal pressureHoop and longitudinal stress derived from two free bodies, plus the sphere. Verified against hoopStress, longitudinalStress and sphereStress. | Covered in fullderivation · 1 worked example | Module 7 · pressure vessels and load sharing |
08Properties of Engineering Materials
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 8.1 | Classification of engineering materialsDuctile, brittle, asymmetric and time-dependent behaviour, each tied to the structural consequence it drives. | Covered in full | Module 8 · the tensile test |
| 8.2 | Testing of engineering materialsTensile test procedure, gauge length and both ductility measures. Numbers verified against sectionProps and axialExtension. | Covered in full1 worked example · interactive | Module 8 · the tensile test |
| 8.3 | Stress–strain curvesNominal vs true stress, necking, and 0.2% proof stress for materials with no yield plateau. | Covered in full1 worked example · interactive | Module 8 · the tensile test |
| 8.4 | Strain hardeningStrain hardening and the ultimate-to-yield ratio as a measure of reserve. | Covered in full1 worked example | Module 8 · the tensile test |
| 8.5 | Creep and relaxationThree stages of creep, and relaxation as the same process seen at constant strain. | Covered in full | Module 8 · creep relaxation fatigue |
| 8.6 | FatigueStress range, stress concentration and detail category. Miner's cumulative damage is implemented and tested but not yet taught. | Covered in full1 worked example | Module 8 · creep relaxation fatigue |
| 8.7 | Design methodsThe source predates current codes. Excluded deliberately: this course teaches mechanics, and any code content must come from current official sources. | Deliberately excluded | Module 8 |
| 8.8 | Material propertiesE, G, nu and K and the relationships between them. Indicative property values only, clearly labelled as not design data. | Covered briefly1 worked example | Module 8 · elastic constants |
09Bending of Beams
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 9.1 | Symmetrical bendingFull first-principles derivation implemented; verified by tests. | Covered in fullderivation · 1 worked example · interactive | Module 8 · deriving bending stress |
| 9.2 | Combined bending and axial loadSuperposition of P/A and My/I, movement of the neutral axis, and the middle-third rule. Verified against sectionProps and bendingStress. | Covered in full1 worked example | Module 9 · combined bending and axial |
| 9.3 | Anticlastic bendingAnticlastic curvature as a Poisson effect. Verified against anticlasticCurvature. | Covered in full | Module 9 · unsymmetrical bending |
| 9.4 | Strain energy in bendingBending strain energy worked through and cross-checked against half-P-delta. Verified against bendingStrainEnergy. | Covered in full1 worked example | Module 15 · work and strain energy |
| 9.5 | Unsymmetrical bendingGeneral bending formula with a product of inertia, and the skewed neutral axis. Verified against unsymmetricalBendingStress. | Covered in fullderivation · 1 worked example | Module 9 · unsymmetrical bending |
| 9.6 | Calculation of section propertiesSecond moment of area and section modulus, with an interactive section explorer. Verified against sectionProps. | Covered in full1 worked example · interactive | Module 9 · second moment and section modulus |
| 9.7 | Principal axes and principal second momentsPrincipal second moments by Mohr's-circle algebra. Verified against principalSecondMoments, including both rotation invariants. | Covered in full1 worked example | Module 9 · unsymmetrical bending |
| 9.8 | Effect of shear on the theory of bendingShear strain warps the cross-section, so plane sections do not strictly remain plane; the span-to-depth limit is given. | Covered in full | Module 10 · shear centre and unsymmetrical sections |
| 9.9 | Load, shear force and bending moment relationships | Covered in fullderivation · 1 worked example · interactive | Module 3 · load shear moment relationships |
| 9.10 | Plastic bendingElastic to fully plastic stress block, equal-area neutral axis, shape factor and the plastic hinge. Verified against plasticModulusRect, shapeFactor, yieldMoment and plasticMoment. | Covered in full1 worked example | Module 9 · plastic bending and hinges |
10Shear of Beams
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 10.1 | Shear stress distribution in unsymmetrical sectionsShear flow traced round a channel, and the shear centre located. Cross-checked against the closed form b2 h2 t / 4I. | Covered in full1 worked example | Module 10 · shear centre and unsymmetrical sections |
| 10.2 | Shear stress distribution in symmetrical sectionstau = VQ/It derived from a beam slice. Verified against rectShearStress, rectMaxShear and iSectionMaxShear. | Covered in fullderivation · 2 worked examples | Module 10 · deriving the shear formula |
| 10.3 | Strain energy due to shearShear strain energy with the form factor k. Verified against shearStrainEnergy. | Covered in full1 worked example | Module 15 · strain energy in shear and torsion |
| 10.4 | Shear in thin-walled open sectionsShear flow traced round an open section from a free edge, and the shear centre located. | Covered in full1 worked example | Module 10 · shear centre and unsymmetrical sections |
| 10.5 | Shear in thin-walled closed sectionsClosed thin-walled shear flow via Bredt, contrasted quantitatively with the open case. Verified against closedThinWalledJ. | Covered in full1 worked example | Module 11 · non circular and open sections in torsion |
11Torsion of Beams
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 11.1 | Torsion of solid and hollow circular barsT/J = tau/r = G.theta/L derived from the geometry of twist. Verified against polarJSolid, polarJHollow, torsionalShearStress and angleOfTwist. | Covered in fullderivation · 2 worked examples | Module 11 · deriving circular torsion |
| 11.2 | Strain energy due to torsionTorsional strain energy, cross-checked against half T theta. Verified against torsionalStrainEnergy. | Covered in full1 worked example | Module 15 · strain energy in shear and torsion |
| 11.3 | Plastic torsion of circular barsYield and fully plastic torques, and the torsional shape factor of 4/3. Verified against torsionYieldTorque and torsionPlasticTorque. | Covered in full1 worked example | Module 11 · torque diagrams and plastic torsion |
| 11.4 | Torsion of thin-walled closed sectionsBredt shear flow q = T/2Am. Verified against bredtShearFlow and bredtShearStress. | Covered in full1 worked example | Module 11 · thin walled and open sections |
| 11.5 | Torsion of solid non-circular sectionsSaint-Venant coefficients for a solid rectangle, and J = (1/3)*sum(b*t3) for thin open sections. Verified against rectangularTorsionStress and openSectionJ. | Covered in full1 worked example | Module 11 · non circular and open sections in torsion |
| 11.6 | Warping of cross-sectionsWarping explained as the reason the circular derivation fails, and uniform torsion distinguished from warping torsion. | Covered in full | Module 11 · non circular and open sections in torsion |
12Composite Beams
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 12.1 | Steel-reinforced timber beamsTransformed section derived from strain compatibility. Verified against transformedSection and compositeStress. | Covered in fullderivation · 1 worked example | Module 12 · strain compatibility and transformed sections |
| 12.2 | Reinforced concrete beamsCracked elastic section, with the neutral axis from bx^2/2 = nAs(d - x). Verified against rcCrackedSection. | Covered in fullderivation · 1 worked example | Module 12 · reinforced concrete sections |
| 12.3 | Steel and concrete composite beamsShear connection, the transformed section and what composite action is worth. Numbers verified against an independent transformed-section calculation. | Covered in full1 worked example | Module 12 · steel concrete composite beams |
13Deflection of Beams
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 13.1 | Differential equation of symmetrical bendingFull Euler–Bernoulli derivation including exact curvature. | Covered in fullderivation · 1 worked example · interactive | Module 9 · euler bernoulli beam theory |
| 13.2 | Singularity (Macaulay) functionsMacaulay brackets, the three rules, and a worked case checked against the standard PL^3/48EI result. | Covered in full1 worked example | Module 13 · macaulay and moment area |
| 13.3 | Moment-area methodBoth moment-area theorems, applied to a cantilever and checked against PL^3/3EI. | Covered in fullderivation · 1 worked example | Module 13 · macaulay and moment area |
| 13.4 | Deflections due to unsymmetrical bendingResolve onto the principal axes, deflect about each, then combine as vectors. | Covered in full | Module 13 · unsymmetrical and shear deflection |
| 13.5 | Moment-area for unsymmetrical bendingMoment-area applied about each principal axis in turn, with the deflections combined as vectors. Verified against deflCantPoint and principalSecondMoments. | Covered in full1 worked example | Module 13 · unsymmetrical and shear deflection |
| 13.6 | Deflection due to shearShear deflection kVL/GA, and the 1/L2 rule for when it matters. Verified against shearDeflectionCantilever. | Covered in full1 worked example | Module 13 · unsymmetrical and shear deflection |
| 13.7 | Statically indeterminate beamsPropped cantilever solved by compatibility, and fixed-ended beams in the following lesson. Verified against proppedCantileverUdl and fixedEndedUdl. | Covered in fullderivation · 1 worked example | Module 16 · indeterminacy and compatibility |
14Complex Stress and Strain
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 14.1 | Representation of stress at a pointThe three components that describe a plane stress state at a point. | Covered in full | Module 14 · stress on inclined planes |
| 14.2 | Stresses on inclined planesTransformation equations derived from equilibrium of a wedge. Verified against stressOnPlane. | Covered in fullderivation · 1 worked example | Module 14 · stress on inclined planes |
| 14.3 | Principal stressesVerified against principalStresses. | Covered in full1 worked example · interactive | Module 14 · principal stresses and mohrs circle |
| 14.4 | Mohr's circle of stressInteractive Mohr's circle lab. Verified against mohrCircle. | Covered in full1 worked example · interactive | Module 14 · principal stresses and mohrs circle |
| 14.5 | Stress trajectoriesTrajectories traced from the extreme fibres to the neutral axis, and related to crack patterns. Verified against principalStresses. | Covered in full1 worked example | Module 14 · stress trajectories |
| 14.6 | Strains on inclined planesStrain transformation equations, and why the shear term carries a factor of one half. Verified against strainOnPlane. | Covered in fullderivation · 1 worked example | Module 14 · strain transformation and mohrs circle of strain |
| 14.7 | Principal strainsVerified against principalStrains and stressFromStrain. | Covered in full1 worked example | Module 14 · strain rosettes and yield criteria |
| 14.8 | Mohr's circle of strainMohr's circle of strain constructed in gamma/2, with the invariants checked. Verified against mohrStrainCircle and principalStrains. | Covered in full1 worked example | Module 14 · strain transformation and mohrs circle of strain |
| 14.9 | Measurement of surface strains (rosettes)45-degree and 60-degree rosettes. Verified against rosette45 and rosette60. | Covered in full1 worked example | Module 14 · strain rosettes and yield criteria |
| 14.10 | Theories of elastic failureTresca and von Mises. Verified against tresca and vonMises. | Covered in full1 worked example | Module 14 · strain rosettes and yield criteria |
15Virtual Work and Energy Methods
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 15.1 | WorkThe half in (1/2)P.delta derived as the area under the load-deflection line. | Covered in fullderivation · 1 worked example | Module 15 · work and strain energy |
| 15.2 | Principle of virtual workEquilibrium set vs compatible set. Verified against unitLoadTrussDeflection. | Covered in fullderivation · 1 worked example | Module 15 · virtual work and the unit load |
| 15.3 | Energy methodsBoth Castigliano theorems, the dummy-load technique, and the link to the unit-load method. Verified against bendingStrainEnergy and castiglianoDeflection. | Covered in fullderivation · 1 worked example | Module 15 · castiglianos theorems |
| 15.4 | Reciprocal theoremsMaxwell and Betti, and why stiffness matrices come out symmetric. | Covered in fullderivation | Module 15 · unit load for beams |
16Statically Indeterminate Structures
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 16.1 | Flexibility and stiffness methodsForce method set out as a five-step procedure. | Covered in fullderivation · 1 worked example | Module 16 · indeterminacy and compatibility |
| 16.2 | Degree of statical indeterminacy | Covered in full | Module 16 · indeterminacy and compatibility |
| 16.3 | Kinematic indeterminacy3j less restraints less axially rigid members, and why it decides between force and stiffness methods. Verified against kinematicIndeterminacy. | Covered in full1 worked example | Module 16 · kinematic indeterminacy and trusses |
| 16.4 | Statically indeterminate beamsPropped cantilever derived from compatibility. Verified against proppedCantileverUdl and fixedEndedUdl. | Covered in fullderivation · 1 worked example | Module 16 · indeterminacy and compatibility |
| 16.5 | Statically indeterminate trussesForce method on a truss with one redundant, cross-checked against an independent compatibility solution. Verified against redundantTrussForce. | Covered in fullderivation · 1 worked example | Module 16 · kinematic indeterminacy and trusses |
| 16.6 | Braced beamsA braced beam as a propped beam whose prop deforms, with both limiting cases. Verified against bracedBeamStrutForce. | Covered in full | Module 16 · portal frames braced beams and two pinned arches |
| 16.7 | Portal framesSymmetric portal by slope-deflection, with sway ruled out by symmetry. Verified against symmetricPortalUdl and independently against slopeDeflection. | Covered in full1 worked example | Module 16 · portal frames braced beams and two pinned arches |
| 16.8 | Two-pinned archesTwo-pinned arch thrust from least work, and why it equals the three-pinned value under a UDL but not under a point load. | Covered in full1 worked example | Module 16 · portal frames braced beams and two pinned arches |
| 16.9 | Slope-deflection methodTwo-span continuous beam solved and verified against slopeDeflection. | Covered in fullderivation · 1 worked example | Module 16 · slope deflection |
| 16.10 | Moment distributionDistribution factors, carry-over and FEMs. Verified against distributionFactors and carryOverFactor. | Covered in fullderivation · 1 worked example | Module 16 · moment distribution and stiffness |
| 16.11 | Introduction to matrix methodsA propped cantilever solved end to end by assembling and reducing the element matrix, agreeing with the force method. | Covered in full1 worked example | Module 16 · moment distribution and stiffness |
17Influence Lines
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 17.1 | Influence lines for beams in contact with the loadVerified against ilReactionLeft and ilMomentAt. | Covered in fullderivation · 1 worked example | Module 17 · what an influence line is |
| 17.2 | Müller-Breslau principleShear jump of exactly 1.0 derived. Verified against ilShearAt. | Covered in fullderivation · 1 worked example | Module 17 · muller breslau |
| 17.3 | Systems of travelling loadsTwo-axle worst position, cross-checked against maxEffectFromLoadTrain. | Covered in fullderivation · 1 worked example | Module 17 · moving loads and load trains |
| 17.4 | Beams not in contact with the loadIndirect loading through cross beams makes the influence line straight between panel points. Verified against ilIndirectLoading. | Covered in full1 worked example | Module 17 · influence lines for trusses and indirect loading |
| 17.5 | Forces in truss membersChord and diagonal influence lines from the method of sections. Verified against ilTrussChord. | Covered in full1 worked example | Module 17 · influence lines for trusses and indirect loading |
| 17.6 | Influence lines for continuous beamsMuller-Breslau on an indeterminate beam, with the influence-line area reproducing the known 3wL/8 prop reaction. Verified against ilProppedCantileverProp. | Covered in full1 worked example | Module 17 · muller breslau |
18Structural Instability
| § | Topic | Status | Where it is covered |
|---|---|---|---|
| 18.1 | Euler theory for slender columnsDerived from EI y″ + Py = 0 as an eigenvalue problem. | Covered in fullderivation · 1 worked example · interactive | Module 10 · deriving euler buckling |
| 18.2 | Limitations of the Euler theory | Covered in full1 worked example · interactive | Module 18 · effective length and limits |
| 18.3 | Failure of columns of any lengthWhy real columns fall below both the squash and Euler loads, and Rankine-Gordon across the range. Verified against rankineGordon. | Covered in full1 worked example | Module 18 · real columns and energy methods |
| 18.4 | Effect of cross-section on buckling | Covered in full1 worked example · interactive | Module 18 · effective length and limits |
| 18.5 | Stability of beams under transverse and axial loadsMoment magnification 1/(1 - P/Pcr), the secant formula, and why superposition fails. Verified against momentMagnification. | Covered in full1 worked example | Module 18 · beam columns |
| 18.6 | Energy method (Rayleigh-Ritz) for buckling loadsRayleigh-Ritz worked with a parabolic trial shape, giving 12EI/L2 against the exact pi^2 EI/L2 — an upper bound, as the theory requires. | Covered in full1 worked example | Module 18 · real columns and energy methods |