Worked Statics Examples

59 solver-verified worked examples — beams, trusses, frames, and more. Each example shows the full equilibrium solution, open and solvable in your browser.

  • Simply Supported Beam Beam · Intro — Classic simply supported beam with a central point load. Introduces reaction forces and beam equilibrium.
  • Cantilever Beam Beam · Basic — Fixed-wall cantilever with a tip point load. Demonstrates wall reactions (force + moment) and internal force distribution.
  • Overhanging Beam Beam · Basic — Beam extending past a simple support with a concentrated load on the overhang. Introduces overhang effects on reaction direction.
  • Triangular Load Beam Beam · Intermediate — Simply supported beam under a linearly varying (triangular) distributed load. Introduces resultant calculation for triangular distributions.
  • Trapezoidal Load Beam Beam · Intermediate — Beam with a trapezoidal distributed load. Requires superposition of uniform and triangular components.
  • Cantilever Canopy UDL Beam · Basic — Fixed cantilever with a uniformly distributed load (UDL) representing a roof canopy. Realistic civil engineering scenario.
  • Midspan Member Moment Beam Beam · Intermediate — Simply supported beam with an applied couple moment at midspan. Introduces internal moment effects at a section.
  • Self-Weight Compound Beam Beam · Advanced — Multi-segment compound beam with self-weight as a distributed load. Demonstrates compound beam analysis and continuity.
  • Three-Bar Truss Truss · Basic — Simple three-member pinned truss. Foundational example for method of joints and truss equilibrium.
  • Warren Bridge Truss Truss · Advanced — Warren truss bridge structure with diagonal members. Realistic highway bridge cross-section analysis.
  • Zero-Force Members Truss Truss · Intermediate — Truss with zero-force members demonstrating member identification rules. Important for efficient analysis.
  • Jib Crane Boom Frame · Intermediate — Classic jib crane with a cable-supported boom and hanging load. Benchmark problem for constraint solving.
  • L-Frame Frame · Intermediate — L-shaped rigid frame with pinned and roller supports. Introduces frame member forces beyond simple beams.
  • Portal Frame Corner Load Frame · Intermediate — Rectangular portal frame with a corner point load. Represents a single bay of a building frame.
  • Rigid Frame with Hinged Prop Frame · Advanced — Frame with an internal hinge (prop). The hinge introduces an additional equilibrium condition.
  • Combined Loading Frame Frame · Advanced — Frame subjected to both distributed loads and point loads. Realistic multi-load condition analysis.
  • Three-Hinged Arch Frame · Advanced — Classic three-hinged arch structure. Introduces arch thrust and the role of the crown hinge.
  • Internal Hinge with Applied Moment Frame · Expert — Arch with an internal hinge and an applied couple moment. Tests moment equilibrium at the hinge section.
  • Gantry Beam with Offset Trolley Beam · Advanced — Overhead gantry crane with a trolley at an eccentric position. Includes self-weight + trolley load.
  • Pulley Hoist Pulley · Intermediate — Fixed-axle pulley hoist with a hanging load. Demonstrates cable tension transmission through a pulley.
  • Ladder Against Smooth Wall Frame · Basic — Ladder leaning against a smooth (frictionless) wall with a rough floor. Classic 2D equilibrium with friction.
  • Rough Support Tipping Post Frame · Intermediate — Post with rough (friction) support tipping under a horizontal force. Illustrates impending motion at rough contact.
  • 8-Panel Symmetric Pratt Roof Truss Truss · Intermediate — 16 m symmetric Pratt roof truss with 7 panel-point loads (40 kN each); teaches Pratt load-path pattern where verticals compress and diagonals tension, and identifies the zero-force apex vertical by bilateral symmetry.
  • 4-Panel Howe Truss Truss · Intermediate — 12 m Howe roof truss with 3 × 30 kN panel-point loads; distinguishes Howe topology from Pratt — diagonals carry compression and verticals carry tension, the opposite of Pratt.
  • K-Truss Panel Truss · Advanced — 8 × 4 m K-truss panel with a 60 kN apex load; the K-junction at midspan forces symmetric diagonal tension while the entire bottom chord is zero-force — teaches load-path concentration in K-topology.
  • Fink Roof Truss Truss · Intermediate — 8 m Fink roof truss with four symmetric 10 kN panel loads; three zero-force members arise under symmetric loading, and the long Fink diagonals carry the panel loads directly to supports.
  • Baltimore Subdivided Bridge Truss Truss · Expert — 16 m Baltimore truss with sub-panel nodes and 7 × 20 kN bottom-chord loads; demonstrates how sub-diagonals relieve outer top-chord panels and produce the characteristic force jump from −10 to −70 kN.
  • Rafter with Distributed Snow Load Distributed Load · Intermediate — Inclined rafter (3-4-5 triangle) carrying a globally horizontal distributed snow load (engine direction="member" maps to global +x, not along-rafter); exposes the counter-intuitive V/M/N diagram that arises when the load direction does not follow the member axis.
  • Inclined Retaining Wall — Lateral Soil Pressure Distributed Load · Intermediate — 4 m rigid post inclined 15° from vertical with a triangular (20 kN/m base to zero tip) horizontal soil-pressure load; the fixed-base moment depends on both post length and inclination via the lateral centroid height.
  • Hip Rafter — Snow Load + Self-Weight Beam · Advanced — 5 m rafter (3-4-5 triangle) carrying both a triangular snow load and uniform self-weight; both load types must be included in section cuts to find the correct peak moment location at s ≈ 2.4 m from the pin.
  • Beam with Partial UDL on Left Half Distributed Load · Basic — 8 m simply supported beam with a 30 N/m UDL covering only the left 4 m; the asymmetric load shifts 75% of the reaction to the pin, and the peak moment at x = 3 m exceeds the value at the load boundary.
  • Haunched Beam — Decreasing Distributed Load Beam · Intermediate — 4 m fixed cantilever with a 1 m unloaded segment at the wall and a trapezoidal load (12 → 4 kN/m) on the outer 3 m; teaches decomposition of a trapezoidal load into uniform + triangular blocks to find reactions.
  • Highway Bridge with Rocker Support Beam · Basic — 12 m bridge beam with three unequal axle loads (60, 90, 45 kN) and a rocker (vertical-only) right-end support; peak moment 427.5 kN·m occurs at the 90 kN axle where shear changes sign.
  • Guided Column — Telescope Leg Frame · Intermediate — Vertical column with a guided base (reacts horizontal force + moment) and a vertical roller at top; a mid-height side load creates a triangular moment diagram and zero axial throughout — demonstrates the guided support moment reaction.
  • Smooth Wall Reaction — Inclined Plank Frame · Basic — 5 m plank (3-4-5 triangle) leaning against a frictionless vertical wall with a worker load at midspan; all vertical load goes to the floor pin, and the frictionless wall provides only a horizontal normal reaction.
  • Suspension Rod — Two Cables Cable · Basic — 6 m horizontal rod suspended from two symmetric 3-4-5 cables with a 40 kN midspan load; symmetric geometry makes the cable forces equal (25 kN each) and unloads the roller, while the rod itself is in tension.
  • Cable-Stayed Pedestrian Bridge Cable · Advanced — Two horizontal bridge beams connected by stay cables to wall anchors; identical cable slopes force equal tensions (25 kN each) and the beams carry pure axial compression — teaches cable-stayed load transfer with internal hinge conditions.
  • Compound Pulley Block-and-Tackle Pulley · Intermediate — Two-beam system with oblique cable constraints forming a block-and-tackle; mechanical advantage is 1.6 (not 2) because the cables are inclined rather than vertical — demonstrates how cable angle reduces effective MA.
  • Gerber Beam Beam · Advanced — Classic two-span Gerber (Gerber-beam) with an internal hinge; the hinge provides an additional equilibrium equation that makes a four-reaction beam statically determinate.
  • Gerber Beam — Three-Span Beam · Advanced — 18 m three-span Gerber beam with two internal hinges and three asymmetric point loads; each hinge imposes a zero-moment condition that yields the five reactions algebraically — teaches multi-hinge Gerber analysis and sign-reversal at interior supports.
  • Compound Arch — Two Internal Hinges Frame · Expert — Fixed-pin arch with two internal hinges, a 90 kN vertical crown load and a 40 kN horizontal side load; all four segments carry compression (arch action), and the fixed wall supplies a 120 kN·m moment reaction.
  • Pitched Portal Frame — Wind + Gravity Frame · Advanced — Pitched portal frame with a 15 kN horizontal wind load at the eave and a 10 kN/m UDL on the windward rafter; combined loading produces a 96 kN·m peak moment in the rafter and zero moment in the leeward column.
  • Angled Bracket — Oblique Member Force Frame · Basic — L-shaped bracket (3 m vertical × 4 m horizontal arm) with a 45° point load at midspan of the horizontal arm; both components of the oblique load contribute to the fixed-base moment — a common student moment-arm error.
  • Inclined Crane Boom — Angled Cable Pull Frame · Intermediate — 45° crane boom pin-supported at base with a mid-boom roller and an oblique cable pull (−60, −100 N) at the tip; both horizontal and vertical cable components contribute to the moment about A — the core teaching trap.
  • Multi-Span Self-Weight Bridge Beam · Intermediate — Three-span bridge (8 + 2 + 4 m) with member self-weights and two 120 N live loads; an internal hinge at the first span junction provides the extra equation — shows that self-weight must be modelled as UDL in section cuts, not as lumped point loads.
  • Steel Frame — Self-Weight + UDL Frame · Advanced — Symmetric rectangular portal frame (4 m columns, 6 m beam) with column self-weights, beam self-weight, and a 10 N/m UDL; symmetric loading splits reactions 380/380 N, and columns carry pure axial compression with zero shear.
  • Crank-Rocker 4-Bar Mechanism Mechanism · Advanced — Grashof crank-rocker linkage (r₁=2, r₂=4, r₃=3.5, r₄=5 m) — satisfies Grashof condition (s+l=7 ≤ p+q=7.5); sweeps the Freudenstein equation through 360° crank rotation to trace the 69.8° rocker oscillation and both toggle positions.
  • Slider-Crank Mechanism Mechanism · Advanced — Centric slider-crank (crank r=1 m, rod L=2.5 m) with a 400 N piston thrust; animates the full 360° stroke, distinguishes kinematic dead centres (θ=0°, 180°) from static singularity angles (θ=90°, 270°), and traces the 2 m piston stroke.
  • Parametric Beam Span Study Beam · Basic — Simply supported beam with a midspan load sweeping both load magnitude and span length; reactions stay equal at P/2 regardless of span, while peak moment grows linearly with span — teaches the span-invariance of symmetric reactions.
  • Parametric Truss Height Truss · Intermediate — Symmetric 5-member truss with a swept apex height h; chord forces scale as 1/h while the vertical web always carries the full 60 kN regardless of h — a direct demonstration of how truss depth controls efficiency.
  • Propped Cantilever — Indeterminate Classifier Beam · Intermediate — Fixed-wall cantilever with an added roller prop under a UDL; introduces statically indeterminate classification (DSI = 1) — the engine correctly refuses to solve and explains why compatibility is required.
  • Applied Couple on Simply Supported Beam Beam · Basic — 8 m simply supported beam with a 24 kN·m CCW couple at x = 3 m; the couple creates no shear but causes a 24 kN·m jump in the moment diagram and forces one reaction downward — a frequently mishandled sign-convention problem.
  • Canopy Panel under Oblique Wind Pressure Distributed Load · Intermediate — 6 m horizontal canopy panel with a 4 kN/m load at 210° (downward-and-leftward wind resultant); both components contribute to reactions, and the axial force is compression — teaches arbitrary-angle distributed load resolution.
  • Cantilever Beam (Imperial Units) Beam · Basic — 15 ft (4.572 m) fixed cantilever with a 5 kip (22.24 kN) tip load; identical physics to the metric cantilever but demonstrates imperial-unit display and the 75 kip·ft fixed-end moment in a realistic engineering context.
  • King-Post Roof Truss Truss · Basic — Inverted king-post truss (tie-beam above, apex below) with a 120 N symmetric central load; the king post is a zero-force member by the collinear-joint rule, and rafters carry tension while the horizontal tie-beam carries compression.
  • Lower Wishbone Control Arm Frame · Intermediate — Triangular automotive wishbone with a pin and vertical roller at the inner pivots and a combined lateral + vertical ball-joint load (20, −110 N); demonstrates truss analysis for a real vehicle suspension component with irrational surd member forces.
  • Zero-Force Members — Isolated-Node Rule Truss · Intermediate — Five-member truss where two members meeting at an isolated unloaded joint D are both zero-force (isolated-node rule); the active load path runs through the triangle only, demonstrating that structural dead-wood carries no force.
  • Rough Support — Sliding Post Frame · Intermediate — Short post on a rough base support with combined lateral (20 N) and vertical (50 N) applied loads; the rough support provides friction + normal + contact moment reactions, and the example checks both sliding onset and tipping eccentricity.
  • Pratt Bridge Truss — Method of Sections Truss · Advanced — Multi-panel Pratt bridge truss solved by method of sections; a single diagonal cut through three members isolates the required forces without solving every joint — the canonical demonstration of sections efficiency over joints.

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