Truss Analysis Calculator

Purpose and model boundary

This calculator analyses a plane, pin-jointed truss drawn on the page: up to 30 joints and 60 members, on pins and rollers (up to 10 supports, at most one per joint), with up to 20 joint loads. For the truss as drawn it returns every member's axial force, in tension or compression, and its stress, the support reactions, the joint displacements, the section properties of each section, and an equilibrium check. Statically determinate and indeterminate trusses are both included. It is a linear-elastic analysis for learning and estimating. It makes no buckling, strength, connection or code check, and it is not a member design: a compression member can buckle at a far lower force than the stress shown suggests.

The spreadsheet is the calculation authority. The page sends the drawn geometry and the values you type to the workbook and draws only what the workbook returns; it computes no engineering value itself.

Inputs and units

Sign conventions: X is to the right and Y is up. Loads, reactions and displacements are positive to the right and up.

Governing relationships

The workbook assembles the stiffness of every member from its length, its area and E, and solves the direct stiffness method for the joint displacements under the joint loads, with the supported directions held. Each member's axial force follows from the change in its length, and the reactions from the forces at the supports.

• A member's stress is its axial force over its area, tension positive.
• The number of reaction components r is two for each pin and one for each roller.
• The count test for stability needs r ≥ 3 and m + r ≥ 2j, for m members and j joints.
• Equilibrium: the sums of the horizontal and vertical forces, and the moment about the origin, of all reactions and loads are zero in exact arithmetic.

Calculation sequence

1. The workbook checks each joint, section, member, support and load row, and refuses the first one that is wrong.
2. Before solving, it tests the truss for stability twice: by the count of members, reaction components and joints (r ≥ 3 and m + r ≥ 2j), and numerically through the stiffness matrix. A truss that passes the count but can still move without stretching a member (an unbraced panel, or supports in line) is refused with NOT VALID, and every result then reads blank.
3. It solves the joint displacements by the direct stiffness method, then each member's axial force and stress and the support reactions.
4. It finds the largest tension, the largest compression, the largest stress and the largest joint displacement, and adds up the equilibrium checks.

Outputs and interpretation

• Members: one line per member row, in the order drawn: its ID, length, area, axial force, whether it is in tension, compression or carries no force, its stress, and its weight when a density is given.
• Reactions: one line per support row: the joint, the support type and its horizontal and vertical reactions.
• Joint displacements: one line per joint row, horizontal and vertical, in inches or millimetres. The largest joint displacement is the size of its horizontal and vertical movement together.
• Largest values: the largest tension, the largest compression, the largest stress (in ksi or MPa) and the largest displacement, each with its member or joint.
• Equilibrium checks: the sums of the horizontal and vertical reactions and loads, and their moment about the origin.

Reading the results:

• Member forces and stresses are positive in tension and negative in compression; the largest compression is the most negative force.
• A roller's reaction in the direction it does not hold reads blank.
• A force or reaction within 1e-9 of the load scale (the sum of every |Fx| and |Fy|) reads 0, with the state Zero; a displacement component within 1e-9 of the largest reads 0.
• On a tie, the largest values name the earliest member or joint row. With no load, they read blank.
• The equilibrium checks are zero in exact arithmetic and read about 1e-15.

On the drawing, each member is coloured by tension or compression as returned, the reactions are drawn as arrows with their sizes, and the deflected shape can be shown exaggerated, with its factor stated.

Validation and status logic

The status reads OK when the truss is solved, or NOT VALID with the first problem to fix: a joint, section, member, support or load row, or a truss that is unstable or a mechanism. Under NOT VALID every number is blank.

Assumptions and limitations

• Linear elastic, pin-jointed plane truss: members carry axial force only, and loads act at the joints.
• No buckling, strength, connection or code check; the stress is not compared with any limit.
• One modulus of elasticity for every member. Section areas are those of the idealized sharp-corner outline.
• The member weights are reported only; they are not applied as a load.
• A value that does not apply, such as a weight without a density, reads blank, never zero.

Restrictions and non-computing states

• E is from 10 to 100,000 ksi; a modulus of zero or less is refused.
• Joint coordinates X and Y are numbers between -1,000 and 1,000 ft.
• A joint load is at most 1,000 kip either way.
• The density is above 0 and at most 1,600 lb/ft³, or blank.
• Every joint is connected to a member, and no two joints are at the same point.
• Each member has a start and an end joint that are both in the joint grid and are not the same joint, does not join the same two joints as another member, and names a section in the sections grid.
• Each load names a joint in the joint grid; each joint has at most one support row; the supports give at least 3 reaction components.

Errors and warnings

Each NOT VALID status names the first problem to fix, for example "the truss is a mechanism: 4 members and 3 reaction components are fewer than twice its 4 joints", "the truss is unstable (a joint or a part can move without stretching a member, or the supports are in line), or its member stiffnesses are too far apart to solve reliably", "node N5 is not connected to any member" or "member row 1 (M1) names a section that is not in the sections grid".

The governing method is the direct stiffness method for a plane truss of pin-ended, axially loaded, linear-elastic members. Each member's stiffness EA/L is set along its direction cosines, the member stiffnesses are assembled by joint, the joint equilibrium equations are solved for the displacements, and the member forces and reactions follow from them. Stability is checked first by the count of members, reaction components and joints. Section areas and second moments are computed from the typed dimensions of idealized sharp-corner outlines; no shape table is used or reproduced.

Sources and further reading

• The direct stiffness method for plane trusses, from the member stiffness matrix to assembly, solution and member forces: A. Kassimali, Matrix Analysis of Structures, 3rd edition, Cengage Learning, 2022, chapter 3, Plane Trusses. Publisher page
• Static determinacy and stability of trusses, the method of joints for checking a determinate truss by hand, and truss analysis by the stiffness method: R. C. Hibbeler, Structural Analysis, 11th edition, Pearson, 2024, chapter 3, Analysis of Statically Determinate Trusses, and chapter 14, Truss Analysis Using the Stiffness Method. Publisher page
• The stiffness analysis of framed structures and the solution of the global equations: W. McGuire, R. H. Gallagher and R. D. Ziemian, Matrix Structural Analysis, 2nd edition, 2000. Free edition from Bucknell University
• Centroids, second moments of area and the parallel-axis theorem, used for the section properties: B. J. Goodno and J. M. Gere, Mechanics of Materials, 9th edition, Cengage Learning, 2018, appendix D, Review of Centroids and Moments of Inertia. Publisher page

No table, chart or text from any of these publications is reproduced, and none of their authors or publishers has reviewed or endorses this calculator.

The workbook was checked against an independent solution of every test case: by the method of joints for the statically determinate cases, and by a separate direct stiffness solution for all of them.

Applicable design codes and a qualified engineer govern real structures.