Truss Solver (Method of Joints)
Finds the force in every member of a plane, pin-jointed truss and the reactions at its supports. Choose a Pratt, Howe or Warren flat truss, or a pitched Pratt or Howe roof truss (with 2 panels, a king post truss). Set the span, height and number of panels, load the top or bottom chord panel points, and add up to two point loads at any joint. The diagram redraws as you type. Students use it to check homework, and designers for first member sizes.
Method
The truss sits on a pin at the left end (B0) and a roller at the right. Every joint is in equilibrium [1], [2]:
ΣFx = 0 ΣFy = 0
With j joints that gives 2j equations, enough for the m member forces and the 3 reactions when m + 3 = 2j. That is what makes a truss statically determinate [1]. A member force is positive in tension and negative in compression.
Joints are named B0, B1… along the bottom chord and T… along the top. Each panel point gets the panel load P. A joint over a support gets P/2, since it carries half a panel. For a uniform load w along the truss, P = w·L/n, with n the number of panels.
In the diagram, members in tension are light blue, members in compression amber and zero-force members dashed grey. Thicker lines carry more force. Loads are amber and reactions light blue.
Limits and common mistakes
- Loads act at the joints only, and the members are weightless two-force members.
- Pratt, Howe and pitched trusses need an even number of panels; the list offers only those.
- A zero-force member is not useless. It braces the compression chord and carries load when the loading changes.
- The compression members must also be checked for buckling, over their own length.
Worked example
A pitched roof truss spans 12 m with a 3 m rise at midspan, in six panels. The roof puts 8 kN on each top chord panel point. Find the reactions and the largest chord forces.
- Choose Truss Type Pitched Howe and Number of Panels 6.
- Enter Span 12 m and Height 3 m.
- Choose Panel Loads Act On Top chord joints and enter Panel Point Load 8 kN. Leave Additional Point Loads at None and press Solve. The Solution shows:
Left Support Reaction (Ay, pin at B0, + up): 24,000 N
Right Support Reaction (By, roller at B6, + up): 24,000 N
Largest Tension: 40,000 N
Largest Compression: -44,721.4 N
Zero-Force Members: 2
Check by hand. The load is 5 × 8 + 2 × 4 = 48 kN, so each reaction is 24 kN. At the support joint the net upward force is 24 − 4 = 20 kN. The top chord slopes at 3 in 6, so sinθ = 1/√5 and its force is 20 × √5 = 44.7 kN. Its horizontal part, 44.7 × 2/√5 = 40 kN, is the bottom chord tension.