Truss Analysis: Method of Joints and Method of Sections
A truss converts bending-dominated behavior into axial-dominated behavior. Where a solid beam under midspan loading develops a bending moment that peaks at the center, a truss routes the same loads through a network of members that carry only tension or compression. Because material at the extreme fiber of a solid section does the most structural work while material near the neutral axis contributes very little, replacing the web with an open framework of axially loaded bars is far more material-efficient for medium to long spans. The price is complexity in fabrication and connections. Understanding how to find the force in every member is the foundation of truss design.
Truss Assumptions and Static Determinacy
Classical truss analysis rests on three idealizations. First, all members are connected by frictionless pins at their ends, so no moment is transferred between members at a joint. Second, loads are applied only at the joints, not along member lengths. Third, members are straight between joints. Under these conditions, every member is a two-force member: the internal force is purely axial, either tension or compression, along the member's longitudinal axis.
Real trusses use gusset plates welded or bolted at joints, not frictionless pins, so some secondary bending does develop due to joint rigidity. Engineers call this secondary stress. For well-proportioned trusses with members whose length-to-depth ratios exceed about 10, secondary stresses are small enough that the pin-joint assumption gives conservative and accurate results for primary design.
A planar truss is statically determinate when it satisfies the relationship m + r = 2j, where m is the number of members, r is the number of support reactions, and j is the number of joints. A simple truss starts with a triangle (3 members, 3 joints) and adds 2 members and 1 joint at each step. A simple roof truss on a pin and roller support has r = 3 (one pin gives two reaction components, one roller gives one). Checking determinacy before starting analysis prevents wasted effort on problems that require different techniques.
Method of Joints: Equilibrium at Every Node
The method of joints applies the two equations of equilibrium in a plane (ΣFx = 0 and ΣFy = 0) at each joint sequentially. At each joint, only two unknown member forces can exist if the method is to work, because only two equilibrium equations are available. Starting at a joint with only two unknown members and working inward solves the entire truss systematically.
The sign convention is critical. Assume all unknown members are in tension (force pointing away from the joint). If the analysis returns a positive value, the member is indeed in tension. A negative result means the member is in compression, and the force magnitude is the absolute value. Writing tension as positive throughout, then checking signs at the end, avoids the confusion of redrawing free body diagrams.
At a joint where three members meet and two are collinear, the third member carries zero force if no external load acts at that joint. These zero-force members appear frequently in trusses designed to be stiff under various loading patterns; they become active when load is applied at intermediate joints under asymmetric conditions. Identifying them early simplifies the analysis by eliminating those unknowns immediately.
Zero-force members are not useless. They prevent buckling of adjacent compression members by reducing their effective length, and they activate under load configurations that differ from the design case. Never remove them from a fabricated truss without analyzing every load case.
Method of Sections: Cutting to Find Specific Forces
When only a few member forces are needed, the method of sections is more efficient than solving the entire joint-by-joint system. The method makes an imaginary cut through the truss, exposing the internal forces in the cut members, and then applies equilibrium to one of the two resulting free body diagrams.
The key rule is that the cut can expose no more than three unknown member forces in a planar truss, because three equilibrium equations are available (ΣFx = 0, ΣFy = 0, and ΣM = 0 about any point). The moment equation is particularly powerful. By choosing the moment center at the intersection of two unknown member forces, those two unknowns drop out and the third is solved directly from a single equation.
Consider a Pratt truss with vertical web members in tension and diagonal web members in compression under downward loading. To find the force in the bottom chord member at midspan, cut through the truss at midspan and take moments about the top chord joint directly above the cut. Only the bottom chord force has a moment arm about that point; the top chord and diagonal forces pass through it. The bottom chord force equals the net moment of external forces on one side divided by the distance between chord centroids. This moment is the equivalent bending moment at midspan, and the chord force divided by the truss depth is the equivalent of M/d, showing the direct connection between beam bending and truss chord forces.
Truss Geometry and Efficiency
The choice of truss geometry affects both structural efficiency and fabrication cost. Common configurations include the Pratt truss (verticals in tension, diagonals in compression under typical downward loading), the Warren truss (diagonal members alternate between tension and compression with no verticals, reducing member count), and the Howe truss (verticals in compression, diagonals in tension, less efficient under gravity load but historically used in timber where compression members in wood are easily spliced).
The parallel chord truss carries uniform load with the largest chord forces at midspan and decreasing forces toward the supports. The pitched or triangular truss (as used in roof structures) has chord forces that vary along the length. In a well-proportioned Pratt roof truss, the maximum chord force occurs near the ridge and at the eave, with the magnitude governed by the horizontal component of the thrust from the sloped top chord.
Panel aspect ratio affects diagonal angles. Diagonals at 45 to 60 degrees to the horizontal are most efficient because they resolve vertical loads into axial force with minimal horizontal component spill. Very shallow diagonals near 20 to 30 degrees carry large forces relative to the applied load, requiring heavier sections. Very steep diagonals create high chord shear transfer demands. Keeping panel geometry near 45 degrees is a good starting point before optimization.
Applying Truss Analysis to Real Design
Once member forces are established by hand or software, design proceeds member by member. Tension members are sized for yielding on the net cross-section area (accounting for holes at bolted connections) and fracture on the effective net area. Compression members must satisfy both yielding and buckling checks, with the effective length depending on end conditions at each joint. Top chord compression members of roof trusses are typically braced out of plane by the roof deck or purlins; bottom chords in tension need less bracing but still require periodic lateral support to prevent overall truss sway.
Connection design at the gusset plates often controls truss economy. A heavily loaded joint may require a large gusset plate with many bolts or a complex weld pattern. Pre-checking gusset geometry against available member depths and panel spacing early in design prevents costly revisions later. The method of joints and sections, applied during preliminary design, gives the engineer a rapid sense of force magnitudes across all members and identifies which connections will be most demanding before detailed analysis begins.