Beam reactions, moment and deflection, plus a triangular truss by joints
Two standard statics solvers in one place, each fenced to the cases it states. The beam solver covers a simply supported beam under a central point load or a uniform load, returning reactions, maximum shear, maximum bending moment and maximum elastic deflection from your span, material stiffness and section size. The truss solver analyses the classic symmetric triangular truss (two top chords, a two-piece bottom chord and a king post) by the method of joints, returning every member force with its tension or compression sign. Both are analysis tools: they tell you what forces a stated structure carries, and the content is plain that whether a member is strong enough is a code-checked design question this page does not answer.
Beam: simply supported
Truss: symmetric triangular, king post
Disclaimer: This calculator gives estimates for general information only. It is not financial, medical or other professional advice. Structural adequacy, member capacity and connections must be designed and checked against the applicable building code by a qualified structural engineer; this page analyses stated ideal structures only.
What this calculator does
The beam half solves the two textbook loadings of a simply supported beam exactly: reactions by equilibrium, maximum shear at the supports, maximum moment at midspan (PL/4 for the point load, wL squared / 8 for the uniform load), and maximum elastic deflection at midspan using your material’s Young’s modulus and the rectangular section’s second moment of area. The truss half solves the symmetric triangular truss joint by joint: support reactions first, then the top chords from the apex geometry, the bottom chord from the support joint, and the king post from the bottom mid joint, reporting each force as tension (pulling away, positive) or compression. Both halves print every intermediate quantity so the analysis can be checked against a hand calculation line by line, which is the entire standard these solvers claim.
How to use it
- Stay inside the stated cases. The beam solver is a simply supported span with the load exactly at midspan or uniformly spread. Loads elsewhere, cantilevers, fixed ends and continuous beams are different formulas; forcing them through these two cases is the classic misuse of beam tables, and the page fences itself rather than pretend otherwise.
- Depth is the lever. For a rectangular section I grows with the cube of depth, so doubling depth cuts deflection eightfold; the calculator makes the trade tangible, and the breadth field exists so the trade is quantified rather than felt.
- Truss signs carry the meaning. Top chords in this geometry are always in compression under downward apex load and bottom chords in tension, the signature of truss action (the structure turns bending into push and pull); the king post carries exactly the bottom mid joint’s load and nothing else, which the second worked example demonstrates.
- Forces are not capacities. A member force of 12.62 kN compression is a demand; whether a given timber or steel piece resists it without buckling depends on length, section, material grade and code rules this page deliberately does not encode, per its brief.
How it is calculated
- Point load at midspan: reactions P/2 each; max shear P/2; max moment PL/4; max deflection P L cubed / (48 E I). Uniform load w: reactions wL/2; max shear wL/2; max moment wL squared / 8; max deflection 5 w L to the fourth / (384 E I). Section: I = b h cubed / 12; E selected or custom in GPa; all unit conversions handled internally (span m, loads kN, section mm).
- Truss (method of joints): reactions (P apex + Q mid) / 2 at each support. Chord length from span and height; top chord force = reaction x chord length / height, compression; bottom chord force = reaction x span / (2 x height), tension; king post force = Q, tension (zero when no mid load). Each is derived at its joint in the analysis text.
Worked examples
Beam, point load. Span 4 m, 10 kN at midspan, steel, 100 x 200 mm section (I = 0.1 x 0.2 cubed / 12 = 6.67e-5 m to the fourth): reactions 5.00 kN each, max moment 10.00 kN m, max deflection 10,000 x 64 / (48 x 200e9 x 6.67e-5) = 1.00 mm. Stiff-looking on paper; the reading section explains why deflection, not strength, is usually the governing check at this scale.
Beam, uniform load. Same beam under 5 kN/m: reactions 10.00 kN, max moment 5 x 16 / 8 = 10.00 kN m again, deflection 5 x 5,000 x 256 / (384 E I) = 1.25 mm: identical peak moment, a quarter more deflection, a comparison that teaches more about load distribution than either case alone.
Truss. Span 6 m, height 2 m, apex load 10 kN, mid load 0: reactions 5.00 kN; chord length 3.61 m; top chords -5 x 3.61 / 2 = 9.01 kN compression each; bottom chords 5 x 6 / 4 = 7.50 kN tension each; king post 0.00 kN, a genuine zero-force member under this loading, which exists for geometry and other load cases rather than for this one. Add a 4 kN mid load and the same joints return reactions 7.00, top chords 12.62 compression, bottom 10.50 tension, king post exactly 4.00 tension.
Reading your results
For beams, read deflection against the serviceability yardsticks designers use (span over 250 or span over 360 are common limits: a 4 m span at 1.00 mm is span over 4,000, extremely stiff; at 16 mm it would be at the span/250 boundary), because floors fail comfort long before they fail strength. For trusses, read the force pattern: chords carry the global bending as a push-pull couple whose size is set by truss depth (double the height halves the chord forces, the geometry dividend the formula shows), and any member the analysis calls zero-force is telling you the load path honestly under this loading only.
Limitations
- Ideal supports, ideal joints. Real supports settle and rotate; real truss joints are plates and bolts with eccentricity, not frictionless pins. The analysis is the standard idealisation every textbook starts from, and engineering judgement prices the difference.
- Self-weight is not included. The beam’s own weight is itself a uniform load; add it to the UDL (density x section area x gravity) when it matters, which for long spans it does.
- Deflection is elastic and instant. Timber creeps under sustained load and concrete cracks and creeps; long-term deflection exceeds the elastic figure by code-specified multiples this page does not apply.
- No capacity, buckling or connection checks. Repeated deliberately: analysis finds demands. Capacity lives in dated code tables and qualified review, exactly as the brief states, and no result here should be read as “this beam is safe”.
Frequently asked questions
Why does deflection depend on E and I when reactions do not?
Because reactions and moments in a simply supported (statically determinate) beam come from equilibrium alone: the structure must balance, whatever it is made of. Deflection is a deformation, so it depends on how stiff the material is (E) and how the section’s material is distributed about the bending axis (I, growing with depth cubed). The same 10 kN load on the same span gives identical moments in steel and timber and deflections roughly eighteen times larger in timber at the same section, which the material selector demonstrates in one click.
What is a zero-force member for, if it carries nothing?
Under the analysed loading it carries nothing; under another (wind uplift, asymmetric snow, a load at a different joint) it may carry plenty, and in all cases it braces the long chords against buckling and holds the geometry during construction. The king post in the worked example is zero-force only because the mid load is zero; hang 4 kN there and it carries exactly 4 kN. Zero-force is a property of a member in a load case, not of the member, and the analysis reports it per case for that reason.
Why do truss chord forces fall as the truss gets taller?
The chords resist the global bending moment as a couple: two equal opposite forces separated by the truss depth, so force = moment / depth. The support-joint arithmetic shows the same result (bottom chord = reaction x span / (2 x height)), and doubling the height halves both chord forces in the calculator exactly as the theory says. Depth is the cheapest strength in structural engineering, which is why trusses and I-beams are tall rather than thick.
Can I use these results to build something?
The analysis is correct for the ideal structures stated, and real building needs the rest of engineering: load derivation from codes (including wind, snow and safety factors), member capacity and buckling checks, connection design, durability and a qualified engineer’s sign-off where the law requires it. This page’s proper role is learning, sanity-checking and preliminary exploration; its disclaimer is not boilerplate, and the brief behind it reserves capacity checks to dated code tables and qualified review.
Related calculators
Method: standard beam formulas for the stated cases and the method of joints for the stated truss geometry, as printed in the method section. Last reviewed: 14 October 2026.