Portal Frame Calculator — Moment, Shear, Axial & Reactions

A portal frame is the workhorse of steel construction — two columns rigidly connected to a beam (flat roof) or to a pair of pitched rafters meeting at a ridge (gable roof). It is the structural skeleton of almost every warehouse, industrial shed, barn, hangar and retail box. A portal frame calculator finds, for a given geometry and loading, the bending moment (M), shear (V) and axial force (N) in every member, the support reactions, and the lateral drift at the eaves.

Why can't you use a beam calculator for this? Because a portal frame is a two-dimensional frame, not a one-dimensional beam. A beam solver models only transverse bending along a single straight member — it has no columns, it cannot carry an axial force, it cannot take a horizontal (wind) load, and it cannot represent inclined rafters. This tool runs a genuine 2D plane-frame finite-element solver: every node carries three degrees of freedom (horizontal u, vertical v, rotation θ), and every member is a frame element with both axial stiffness EA and flexural stiffness EI. That is the same direct-stiffness method used by commercial frame software — assembled, constrained by the base fixities, and solved as K·u = F by Gaussian elimination with partial pivoting.

Because the frame is statically indeterminate (a fixed-base portal is redundant to the third degree, a pinned-base portal to the first), there is no single textbook formula that covers it — the moment distribution depends on the geometry, the base fixity and the relative member stiffnesses. Simple formula tools simply cannot do it; SkyCiv's free tier famously caps you at three members, which is one short of a complete portal. Here you get the whole frame — four members, gravity plus wind, both design codes — for free, with the M/V/N diagrams drawn directly on the frame the way a structural engineer sketches them on the drafting board.

1. Choose the roof and the bases. Gable (pitched) builds two inclined rafters and a ridge; Flat beam roof builds a horizontal beam between the columns. Fixed bases clamp the column feet (moment-resisting foundations); Pinned bases let them rotate freely (holding-down bolts only). The support symbols in the sketch follow the standard conventions — hatched clamp for fixed, triangle for pinned.
2. Set the geometry — span L (column to column), eaves height h, and, for a gable, the apex rise f above the eaves. Everything is drawn to scale and dimensioned as you type.
3. Apply the loads. Enter the gravity load w (dead + live + snow) as a downward UDL on the roof members — drawn as a vertical arrow band onto the rafters. For the wind, choose the source: Manual H to type the horizontal eaves force directly (positive pushes to the right; a negative value is suction from the other side), or NBR 6123 to derive it — enter the basic wind speed V₀, the terrain roughness category and the bay spacing, and the real NBR 6123 velocity-pressure engine returns q, Vk, S₂, the wall/roof external pressure coefficients Cpe and the resulting eaves force H. Tick Self-weight to add the selected profile's kg/m to the roof load in one click.
4. Pick the section and fy. Choose an I / H profile from the catalog; the yield strength fy drives the strength check.
5. Read the results — they update instantly, no “calculate” button. The KPI strip shows the maximum moment, axial, shear, the lateral drift and the governing utilization. The M, V and N diagrams are plotted on the frame geometry with the global maxima annotated, and the solved base reactions (horizontal, vertical, and the clamping moment on fixed bases) are drawn as arrows on the sketch.
6. Pick the combination code. Choose the ULS load-combination standard (NBR 8800, AISC 360, EN 1993, AS 4100, IS 800 or NBR 14762). The calculator solves the gravity and wind load cases separately and, because the frame is linear, superposes them under that code's codified γ factors (1.4G, G+1.4W, 0.9G+1.4W for uplift, …) to build a proper ULS envelope — the governing combination is highlighted. This is a real combination check, not a single hand-entered load case.
7. Compare the codes. NBR 8800 and AISC 360 are evaluated side by side on the envelope demand — the combined axial-plus-bending interaction (H1-1) and the elastic stress at the governing section. Click either card to make that resistance code govern the ranking.
8. Read the stability flags. A buckling screening surfaces the column effective-length slenderness (λ = K·h/rx vs the KL/r ≤ 200 cap and the Euler load N,cr) and the rafter lateral-torsional-buckling flag (unbraced length L,b vs the limiting L,p) — the caveats the strength check excludes — routing you to the full member verification in the 3D editor.
9. Check the lightest sections that pass table — the whole flexural catalog is screened against the envelope demand and the five lightest sections that satisfy the check are ranked. Click Use to adopt one.
10. Export or share for free — a CSV of every N/V/M sample plus the reactions, and Copy link to this frame produces a permalink (carrying the geometry, loads, section, wind source and combination code) that rebuilds your exact model for a colleague. Open this portal in the 3D editor carries the same encoded state so the editor can rebuild THIS frame instead of a blank scene.

Tip: the SI ⇄ imperial toggle converts every input and output (kN ↔ kip, m ↔ ft, kN/m ↔ kip/ft, MPa ↔ ksi) — the math always runs in SI internally.

The solver does not evaluate these formulas — it solves the stiffness system numerically — but for the classic cases its output matches them, which is the way to validate any frame tool. For a rectangular portal, pinned bases, under a horizontal load H applied at the beam level (height h, span L):

For gravity on the roof the response is fully indeterminate and geometry-dependent — the columns pick up an inward thrust and end moments, and the beam/rafter midspan moment falls between wL²/8 (as if simply supported, pinned bases, flexible columns) and wL²/12 (as if fully clamped) depending on the base fixity and the column-to-beam stiffness ratio. Two useful bounds the solver honours exactly by equilibrium:

• Total vertical reaction = total applied gravity = w · (roof member length) — for a flat roof that is w·L; for a gable it is w times the summed rafter length 2·√((L/2)² + f²).
• Total horizontal reaction = − applied wind H (the frame is in global equilibrium: ΣFx = ΣFy = ΣM = 0).

After the analysis, the design stress and the checks at the governing section are:

• Combined elastic stress: σ = N/A + M/Sx (axial plus strong-axis bending).

• Combined resistance, both codes side by side: the AISC 360 H1-1 (and equivalent NBR 8800 §5.5.1.2) interaction, with Pr = |N|, Mr = |M|, Pc = φ·A·fy, Mc = φ·Zx·fy:

  if Pr/Pc ≥ 0.2: Pr/Pc + (8/9)·Mr/Mc ≤ 1 · else: Pr/(2Pc) + Mr/Mc ≤ 1

  NBR 8800 uses φ → 1/γa1 (γa1 = 1.10) where AISC uses φ = 0.90. The calculator shows both utilizations in parallel.

This is a first-order strength screening — it deliberately excludes column flexural buckling and rafter lateral-torsional buckling, which need the effective lengths and the full member verification the 3D editor runs.

• Geometry: x to the right, y up, origin at the left column base. Corner nodes: left base, left eaves, apex (gable only), right eaves, right base.
• Loads: the roof gravity w is a downward UDL applied per unit member length (for a flat roof this equals the load per horizontal metre; for a gable, multiply by cos(pitch) if your load is defined per plan area). The wind H is a horizontal point load at the left eaves node, positive to the right.
• Members: each column and each rafter is meshed into 12 Euler-Bernoulli frame elements (6×6 local stiffness with axial + bending terms), rotated to global coordinates by the standard transformation and assembled into the global K.
• Supports: pinned fixes the two translations (u, v) and leaves the rotation free → zero base moment; fixed also fixes the rotation → a clamping moment reaction. Applied by direct elimination of the constrained DOFs.
• Internal forces: axial N positive in tension; shear V and bending moment M recovered from the element end-forces (S = k·u_local − f_equivalent). The moment diagram is drawn offset perpendicular to each member's centreline so the shape reads directly on the frame.
• Reactions: horizontal (Rx), vertical (Ry, positive up) and — on fixed bases — the clamping moment (Mz), drawn as arrows with their values at the two feet.
• Drift: the horizontal displacement of the left eaves node, reported in mm (the classic serviceability check is drift ≤ h/300 … h/150 for wind).

The engine is a direct-stiffness (matrix) plane-frame solver — the same method as commercial software, not a lookup table:

• 3 DOFs per node (u, v, θ); 12 frame elements per member; consistent nodal loads for the gravity UDL (transformed into the member's local axial + transverse components, exact for a uniform load);
• boundary conditions imposed by elimination; K·u = F solved by Gaussian elimination with partial pivoting; base reactions recovered as R = K_row·u − F.

Verification (2026-07-12). For a pinned-base rectangular portal with a horizontal load H at the eaves the solver reproduces the closed form to within the physical axial flexibility of the beam: with H = 20 kN, h = 5 m, L = 8 m it returns base horizontal = 10.00 kN (= H/2), base vertical couple = 12.50 kN (= H·h/L), and eaves moment = 50.02 kN·m against the axially-rigid theory value of 50.00 kN·m (a 0.04% difference — the theory assumes an inextensible beam; the FEM includes the real EA). Global equilibrium ΣFx, ΣFy and ΣM close to machine precision for every case, including the gable frame under combined gravity and wind.

Assumptions: linear-elastic material (E = 200 GPa), small displacements (first-order — no P-Δ), prismatic members (constant EA, EI along each member, all members the same profile), rigid (moment-resisting) eaves and ridge connections, loads in the plane of the frame, and out-of-plane restraint provided by purlins/girts and bracing. A key consequence of the single-profile assumption: because a uniform scaling of every member's stiffness leaves the internal forces of an indeterminate frame unchanged, the N/V/M demand is independent of which profile you pick — only the drift and the utilization change — which is exactly why the "lightest section that passes" ranking can screen the whole catalog against one demand.