How a Truss Bridge Holds: Finite Element Analysis of Tension and Compression

Why triangles, and why pin joints matter

A truss is a structure built entirely from straight members connected at joints, arranged so every panel is a triangle. Triangles are the only polygon that cannot change shape without changing the length of a side — a rectangle can rack into a parallelogram, a triangle simply cannot deform without a member stretching or compressing. Classical truss theory takes this further with two idealising assumptions: every joint behaves as a frictionless pin (it transmits force but no bending moment) and every external load is applied only at joints, never along a member's length. Under those assumptions each member becomes a two-force member — the only way it can be in equilibrium with force applied solely at its two end pins is a single force acting along its own axis, meaning every member carries pure tension (being stretched) or pure compression (being squeezed), and never bends.

Warren, Pratt and Howe: three ways to triangulate a span

The three classic bridge trusses arrange the same triangulated idea with different practical trade-offs. A Warren truss uses a repeating zig-zag of equal-length diagonals with no vertical members at all — the simplest, most material-efficient layout for a given span, though it needs longer unsupported diagonal lengths. A Pratt truss adds vertical members and orients its diagonals so they slope toward the centre, which under typical gravity loading puts the diagonals in tension — historically valuable because tension members could be thin steel rods instead of bulky compression-resistant beams. A Howe truss mirrors the diagonal direction so they carry compression instead, which made more sense in the era of timber construction, since wood resists compression well but timber joints handle tension poorly.

The stiffness-matrix method

For a small truss you can solve every joint by hand with the method of joints — sum the forces to zero at each pin, one joint at a time. A real bridge with dozens of members needs something more systematic: the finite element method, applied here in its simplest form, the direct stiffness method. Each member is treated as a 1D "finite element" whose stiffness (how much force it takes to stretch it by a given amount) depends on its cross-sectional area, its material's Young's modulus, and its length. Every member's individual stiffness contribution is assembled — rotated into global x/y coordinates and summed at shared joints — into one large global stiffness matrix that captures the whole structure at once.

K · u = F

K = global stiffness matrix   (assembled from every member's EA/L, rotated
                                 into the shared x,y joint coordinate system)
u = unknown joint displacements  (x,y shift of every free joint)
F = applied loads at each joint  (traffic load, self-weight, etc.)

solve once for u  →  member elongation = (change in length along its axis)
member force = (E·A / L) · elongation     (Hooke's law, tension + or compression −)

Solving that one matrix equation gives the displacement of every joint simultaneously, and from those displacements each member's elongation — and therefore its exact tension or compression force — follows directly from Hooke's law. This is why the truss visualiser can colour every member by its force the instant you change the topology or the load: the whole structure is one linear system, solved once, rather than dozens of separate hand calculations.

Reading the colours: tension, compression and failure

Once member forces are known, comparing each one against the material's allowable tensile and compressive stress tells you whether a design is safe. Tension failure is comparatively simple — a member in tension either holds or snaps. Compression is more subtle: a slender member can fail by buckling — suddenly bowing sideways — at a load far below what its material could otherwise withstand in pure crushing, which is why compression members in real bridges are typically made thicker or braced, while tension members can stay slender.

Frequently asked questions

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