A truss is a structural framework of slender members (bars) arranged in triangular units, where every bar carries only axial load — pure tension or pure compression — with no bending. This elegant simplification makes trusses both highly material-efficient and mathematically tractable using the method of joints (equilibrium at each pin) or the method of sections (free-body diagrams through cut members). The Pratt truss, patented in 1844 by Thomas and Caleb Pratt, places vertical members in compression and diagonal members in tension — an arrangement that minimises material in long-span steel bridges because steel is far cheaper in tension than in compression (no risk of buckling).
This simulator solves truss equilibrium using the direct stiffness method (a form of the finite element method), assembling a global stiffness matrix from member stiffnesses and solving for joint displacements under applied loads. Members are coloured blue when in tension and red when in compression, with colour intensity proportional to force magnitude. You can adjust span, panel height, number of panels, applied load position, and material cross-sectional area to explore how geometry and loading affect force distribution and critical member stresses.
What is the method of joints for solving a truss?
The method of joints applies equilibrium (ΣFx = 0, ΣFy = 0) at each pin joint in sequence. Starting at a joint with only two unknown member forces, you can solve for both. Then move to an adjacent joint where the solved force is now known, and repeat until all forces are found. A truss with j joints and m members is statically determinate (uniquely solvable) if m = 2j − 3; more members produce redundant (statically indeterminate) structures requiring compatibility equations.
Why are diagonal members in a Pratt truss in tension while verticals are in compression?
Under a central downward load, the bottom chord is in tension and top chord in compression (like a beam). The vertical members carry the vertical shear forces and are placed in compression (shorter members buckle at higher loads than longer ones). The diagonals resist shear by going into tension — an efficient arrangement because steel wire or rods can carry tension indefinitely without buckling, whereas long compression members need larger cross-sections to avoid Euler buckling. This is why Pratt trusses are preferred for medium-span railway bridges.
What is the direct stiffness method used in this simulator?
Each bar element has a local stiffness k = EA/L (axial rigidity), where E is Young's modulus, A is cross-sectional area, and L is length. The element stiffness matrix is rotated from local (axial) to global (x-y) coordinates using a transformation matrix. Element matrices are assembled into a global stiffness matrix K, boundary conditions are applied (pinned and roller supports constrain certain displacements to zero), and the system Kd = f is solved for joint displacements d. Member forces follow from F = EA/L × elongation.
A long slender bar in compression can buckle laterally before reaching its material yield strength. The critical Euler buckling load is Pcr = π²EI/Leff², where I is the second moment of area and Leff is the effective length (= L for pin-pin, L/2 for fixed-fixed). The slenderness ratio L/r (r = radius of gyration = √(I/A)) governs buckling: members with L/r > 120 are highly slender and heavily prone to buckling. Bridge designers select wider cross-sections or add lateral bracing for compression members to raise Pcr well above the design load.
Pratt truss (1844): vertical members in compression, inclined members in tension — efficient in steel, common in railway bridges. Howe truss (1840): vertical members in tension (iron rods), inclined members in compression — suited to timber diagonals supporting iron rods. Warren truss (1848): equilateral triangles with no vertical members; alternating diagonals carry tension and compression depending on load position. The Warren truss uses fewer members for the same span and is popular in motorway flyovers and long-span pedestrian bridges because it is stiffer against lateral wind loads.
The most common failure modes are: (1) member buckling (slender compression chord in a long-span bridge); (2) fatigue fracture at pin connections from millions of repeated traffic load cycles (failure mode in the 2007 Minneapolis I-35W bridge collapse); (3) corrosion reducing cross-sectional area, particularly in older riveted bridges; and (4) overload from flooding or vehicle overweight. Modern bridge codes (Eurocode EN 1993-2; AASHTO LRFD) apply load factors and resistance factors to maintain design reliability indices β ≥ 3.5.
The method of sections cuts the entire truss with an imaginary plane through three members and applies equilibrium to one of the two resulting free-body diagrams. This finds the forces in exactly those three cut members without solving the rest of the truss — ideal when you need the force in just one critical member (e.g., the bottom chord under a train). For a long truss, starting from one end with the method of joints and working inward is often more systematic, while the method of sections is faster for checking specific members mid-span.
A statically indeterminate (redundant) bridge has more members than the minimum needed for equilibrium. Extra members provide alternative load paths: if one member fails, forces redistribute to neighbours rather than the structure collapsing. Modern bridge codes classify bridges as fracture-critical (non-redundant — failure of one member causes collapse) or not. Single-box girder bridges are fracture-critical; multi-girder bridges are not. The FHWA requires fracture-critical bridges to undergo more frequent inspections (every 2 years vs 4 years for non-critical structures).
Dynamic loads cause resonance when the load frequency matches a natural frequency of the structure. The Tacoma Narrows Bridge (1940) collapsed due to aeroelastic flutter — torsional oscillation driven by wind vortex shedding at a frequency near the bridge's torsional mode. Modern bridge aerodynamics uses section model wind tunnel testing to verify flutter speed is well above the design wind speed. For traffic, dynamic amplification factors (1.1–1.4× the static load) are applied in codes to account for vehicle bounce and road surface roughness excitation.
Structural steel (S275, S355, or high-strength S690 to EN 10025) is the standard material for large truss bridges, offering a good strength-to-weight ratio (tensile strength 430–770 MPa) and ease of fabrication and welding. Weathering steel (Corten, S355J2W) forms a stable oxide layer that eliminates painting. Aluminium alloys (6082-T6, 7075) are used for lightweight pedestrian bridges and military pontoon bridges. Fibre-reinforced polymer (FRP) pultruded sections are increasingly used for bridge decks in highly corrosive environments (coastal bridges, chemical plant walkways).
The Quebec Bridge (Canada, completed 1919) holds the record for the longest cantilever truss span at 549 m between anchor arms. Its construction was marked by two catastrophic collapses (1907 and 1916) due to inadequate design of the lower chord compression members, killing 89 workers. The Firth of Forth Bridge (Scotland, 1890) has a cantilever span of 521 m and was the world's longest bridge span of any type for 27 years. For pure truss spans, the Ikitsuki Bridge (Japan, 1991) has a continuous truss span of 400 m.