Truss Calculator

How is a truss solved?

A truss is a structure of straight members joined by pins at their ends. When the loads act at the joints, each member carries axial force only: tension or compression. This calculator takes the joint coordinates, members, supports and joint loads and uses the direct stiffness method to find the force in every member, the support reactions and the joint displacements. In the drawing, tension and compression members have different colours and a thickness that follows the force.

Method

The stiffness of a member is k = E × A / L. A 4 × 4 element matrix is built from the direction cosines of the member, all members are assembled into one system matrix, the supported degrees of freedom are separated and K × u = F is solved. The member force follows from the difference between its end displacements: N = (E × A / L) × (Δu × cos θ + Δv × sin θ). A positive value is tension and a negative value is compression. Reactions come from R = K × u − F and are checked against equilibrium of the whole structure.

Static determinacy

Let m be the number of members, r the number of reaction components and j the number of joints. If m + r = 2 × j the truss is statically determinate and the member forces do not depend on the cross-sectional areas. If m + r is larger the truss is indeterminate and the forces depend on the relative stiffness of the members. If it is smaller the structure is a mechanism. Even when the count works out, a panel without a diagonal makes the truss unstable. The calculator detects this from a singular stiffness matrix and tells you which joint to look at.

Example

Take a four panel Pratt truss with 3 m panels and a height of 4 m, loaded with 20 kN at each bottom chord joint. Each reaction is 30 kN. The inclined end member carries 30 / 0.8 = 37.5 kN in compression, the bottom chord 22.5 kN in tension, the top chord 30 kN in compression and the diagonals 12.5 kN in tension. The middle vertical is a zero force member. Switching the same joints to a Howe layout turns the diagonals into 12.5 kN compression members.

Limits

Loads act only at the joints, so loads along a member and member self weight are not included. Joints are treated as frictionless pins. Buckling of compression members, connections and resistance checks are outside this tool. Inclined rollers and support settlement cannot be defined. The result is a preliminary calculation.