Worked examples

Learn structural analysis

35 textbook examples, each solved in full by hand with the classical method — then checked against this site’s solver to round-off, and one click from being live in the calculator. Read the derivation, see the diagrams, then hit Try it to open the exact model and change anything you like.

35
worked examples
10+10+10+5
beam · truss · frame · section
100%
hand-verified

Beam worked examples

01

Simply supported beam with a UDL

The first beam every engineer solves — wL²/8, and where the classic formulas come from.

FoundationalDeterminateUDLDeflection
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02

Simply supported beam with a point load

The other first-principles beam: Pab/L, and why the shear diagram is two flat steps.

FoundationalDeterminatePoint load
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03

Cantilever with a tip load and a UDL

Fixed-end behavior: the hogging moment, and superposing PL³/3EI with wL⁴/8EI.

FoundationalCantileverPoint + UDLSuperposition
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04

Simply supported beam with a triangular load

A load that grows along the span: thirds at the supports, and the peak at L/√3.

IntermediateDeterminateTriangular load
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05

Overhanging beam with a partial UDL and an applied moment

Contraflexure: where sagging turns to hogging — and why an end moment travels the overhang.

IntermediateOverhangStaticsContraflexure
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06

A drop-in span on a shear connection

The hinge that turns two spans back into statics — and the fin plate that builds it on site.

IntermediateGerberShear connectionStatics
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07

Two-span continuous beam

Indeterminacy: the hogging moment over the middle support, by the three-moment theorem.

AdvancedIndeterminateThree-momentContinuous
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08

Support settlement in a continuous beam

Why a settling support bends an indeterminate beam — real moments with zero applied load.

AdvancedIndeterminateSettlementCompatibility
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09

A welded moment splice in a continuous beam

A full-strength welded splice keeps the beam continuous — and the I step moves deflections, not moments.

AdvancedIndeterminateWelded connectionStepped sections
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10

A suspended-span bridge on two shear connections

Gerber at full scale: two hinges make four supports solvable by statics alone.

AdvancedGerberShear connectionBridge
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Truss worked examples

11

The simplest truss: one triangle

Three members, one load — where tension and compression come from.

FoundationalDeterminateFirst principles
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12

Warren truss by the method of joints

The classic triangulated form: alternating tension–compression diagonals, joint by joint.

IntermediateDeterminateMethod of joints
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13

Fink roof truss under gravity loads

The everyday roof truss: rafters in compression, the bottom chord tying them in tension.

IntermediateRoofMethod of joints
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14

Howe truss: the Pratt inverted

Slope the diagonals the other way and every one flips into compression.

IntermediateDeterminateMethod of jointsBridge
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15

A fink truss with gusset-plate joints

Reading the joints: where gussets go, and why the analysis still treats every joint as a pin.

IntermediateRoofGusset platesMethod of joints
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16

A cantilever jib truss on a wall

Cantilever anatomy: the tie, the strut, the zero-force member — and the anchor that feels it all.

IntermediateCantileverZero-force member
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17

Pratt through-bridge truss

How a bridge carries deck loads: chords take the moment, diagonals take the shear.

AdvancedMethod of sectionsBridgeDeterminate
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18

A settled pier in a continuous truss

Settlement forces an indeterminate truss — and leaves a determinate one untouched.

AdvancedIndeterminateSettlementFlexibility method
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19

A member fabricated short: lack of fit

Forcing a too-short member into a redundant truss locks in stress — with zero applied load.

AdvancedSelf-stressLack of fitVirtual work
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20

A heated member in a redundant truss

Temperature is a load: heat one member of a redundant truss and the whole frame fights back.

AdvancedSelf-stressThermalVirtual work
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Frame worked examples

21

Pinned-base portal frame under gravity load

The frame that industrial sheds are made of — why rigid joints create thrust a truss never has.

FoundationalPortalIndeterminateUDL
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22

Fixed-base portal frame — what base fixity buys and costs

The same frame with fixed feet: stiffer, less sway, and a moment the footing now has to carry.

IntermediatePortalIndeterminateBase fixity
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23

Portal frame under lateral wind — sway and the windward/leeward split

Why a lateral load makes one column work far harder than the other, and where sway comes from.

IntermediatePortalWindSway
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24

Portal frame under combined gravity and wind

Why the governing case is a combination, and how gravity can rescue a base from uplift.

IntermediatePortalLoad combinationWind
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25

Two-story frame — how story shear accumulates downward

Why the ground-floor columns of a building are always the hardest working ones.

AdvancedMulti-storyWindStory shear
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26

Two-bay portal frame — what the interior column really carries

Why an interior column takes twice the axial load but a fraction of the moment.

AdvancedMulti-bayContinuityUDL
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27

Pitched-roof portal frame — the real industrial shed

What a roof pitch does to thrust, and why the apex is a moment connection worth detailing.

AdvancedPortalPitched roofInclined members
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28

Moment vs shear connections — what a release actually costs

Release the rafter ends and the frame changes character — the connection IS the structure.

IntermediateConnectionsMoment releasePortal
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29

Support settlement in a frame — stress from movement alone

An indeterminate frame develops real forces when a footing settles, with no load applied at all.

AdvancedSettlementIndeterminateServiceability
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30

An applied moment at the eaves — crane brackets and eccentric connections

Where an eccentric load really goes, and why a bracket is a moment you must chase to the ground.

IntermediateApplied momentEccentricityCrane
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Cross-section worked examples

31

Adding a cover plate — where the extra steel actually pays

Steel far from the centroid earns its keep by d²: a plate on one flange only, and why Zx top and bottom diverge.

FoundationalBuilt-upSection modulus
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32

60E1 railway rail — properties of a real published profile

A monosymmetric section: why the centroid is not at mid-height, and what that does to the two section moduli.

IntermediateMonosymmetricPublished profile
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33

Welded box girder — what closing a section buys you

A closed box against an open I of the same depth and weight: Iy multiplies, and that is what resists twist and buckling.

IntermediateHollowBox girder
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34

Composite bridge deck — a concrete slab on a steel plate girder

The transformed-section method: one modular ratio turns two materials into one section, and triples the stiffness.

AdvancedCompositeTransformed section
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35

Unequal angle — why it bends the way it does, not the way you pushed it

A non-zero product of inertia rotates the principal axes: load an angle vertically and it deflects sideways.

AdvancedPrincipal axesIxy
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Guides & theory

Start here

The references — every claim illustrated

Starting from scratch

Frequently asked questions

Are the worked examples on StructureCalcs verified or trustworthy?
Yes. Every worked example is solved in full by hand with the classical method and then checked against the site’s own Rust/WASM solver, with the hand value and the solver value shown side by side in a verification table on every example page, and the two agree to round-off. That agreement is backed by an end-to-end harness that runs every example through the real engine, plus a unit-test suite that pins every preset and its hand arithmetic. These checks are run by the maintainers to keep the numbers honest, rather than automatically gating each production build.
How many worked structural analysis examples does StructureCalcs have?
The Learn section has 35 hand-verified worked examples: 10 beam, 10 truss, 10 frame and 5 section examples. They are grouped by tool on the Learn index and ordered from foundational to advanced within each group.
What structural analysis methods do the StructureCalcs worked examples cover?
The beam examples cover statics and superposition, the three-moment (Clapeyron) theorem for continuous indeterminate beams, compatibility for support settlement, and Gerber internal hinges. The truss examples cover the method of joints, the method of sections, zero-force members, and the flexibility/virtual-work method for settlement, lack-of-fit, and thermal effects. The frame examples cover portal frames pinned and fixed at the base, sway and story shear, moment releases, inclined and pitched members, and load combination. The section examples cover built-up and composite sections, the transformed-section method, principal axes and the product of inertia. Each example is tagged Foundational, Intermediate, or Advanced.
Can I open a StructureCalcs worked example in the calculator and change it?
Yes. Every worked example has a Try it button that loads the exact model straight into the Beam or Truss calculator, where you can edit any input and re-solve. The same preset can also be deep-linked directly with a /beam?example=<id> or /truss?example=<id> URL.
Are StructureCalcs trusses actually pin-jointed even though the drawings show gusset plates?
Yes. The truss drawings may show gusset plates or bolts at the joints for realism, but the analysis idealises every joint as a frictionless pin, exactly as the method of joints requires — drawn as built, analyzed as pinned. The Learn examples say this explicitly (the fink-with-gussets and simple-triangle pages spell it out), and welded truss joints were deliberately removed because a pinned solver cannot honestly claim moment continuity.
Does StructureCalcs have worked examples for support settlement, thermal effects, and fabrication errors?
Yes. The truss set includes a settled-pier example solved by the flexibility method, a lack-of-fit example where one diagonal is fabricated 5 mm short, and a thermal self-stress example where a member is heated in a redundant frame — all producing real member forces with zero applied load. The beam set also includes a support-settlement example solved by compatibility, and the examples note that downward settlements are entered as negative values.
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