Solve a single-bay, single-storey rigid frame (portal frame) under a lateral load with the approximate portal method. Adjust the column height, beam span, lateral load and flexural stiffness to see the column shear, bending moment, axial force, beam end moment and storey drift update in real time.
Parameters
Column height H
m
Storey height from the base to the beam level
Beam span L
m
Centre-to-centre distance of the two columns (bay width)
Lateral load P
kN
Horizontal force at the beam level (wind, earthquake, etc.)
Column flexural stiffness EI
kN·m²
Flexural stiffness of the column section. Governs storey drift
Results
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Shear per column (kN)
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Column bending moment (kN·m)
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Column axial force (kN)
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Beam end moment (kN·m)
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Beam shear (kN)
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Storey drift (mm)
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Portal frame and lateral sway — animation
A single-bay frame of two columns and a beam joined by rigid corners sways sideways under the lateral load P at beam level. The white dots mark the inflection points at the column mid-heights and beam mid-span; the base arrows are the support reactions.
Shear per column V_col, bending moment at the top and base M_col, and column axial force N_col. P: lateral load, H: column height, L: beam span. The portal method assumes inflection points at the mid-height of columns and the mid-span of the beam.
Beam end moment M_beam (equal to the column top moment by joint equilibrium) and beam shear V_beam.
$$\delta=\frac{P\,H^{3}}{24\,EI}$$
Storey drift δ (lateral sway). The inflection point at each column mid-height gives the two columns a storey stiffness of 24EI/H³. EI: column flexural stiffness.
What is a Portal Frame?
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A "portal frame" is a skeleton with two columns and a beam on top, shaped like a doorway, right? What makes it special?
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Exactly — a doorway-shaped frame. It is the workhorse unit of building and industrial-shed steelwork. The key point is that the columns and beam are joined by rigid connections, ones that also transfer moment. With pin connections the frame would simply topple sideways, but with rigid joints the two columns and the beam act as one continuous structural unit and can resist sideways loads. That is why you can leave the bay completely open and usable, with no diagonal bracing.
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No bracing needed — that's handy. But how does it actually resist a horizontal force? The columns don't just prop it up, do they?
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Good question. When a lateral load P acts at beam level, the columns first pick up a "shear". In a single bay the two columns each carry half of P, so the shear per column is P/2. That shear bends the columns, producing a "bending moment" at the top and base of each column. On top of that there is an overturning moment P·H trying to tip the whole frame over. That is resisted by axial forces — the windward column gets pulled in tension, the leeward column gets pushed in compression, a couple. Raise P with the slider on the left and you will see the shear, the moments and the axial force all grow together.
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I learned that a rigid frame is statically indeterminate. So how can you find the member forces from equilibrium alone?
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Strictly you cannot — and that is exactly why we use the "portal method", an approximate analysis. The portal method makes two assumptions. One: there is an inflection point at the mid-height of every column and the mid-span of the beam. An inflection point is where the bending moment passes through zero. Under a lateral load the members bend into an S-shape (double curvature), so the moment reverses near the middle — that assumption is physically reasonable. Two: in a single bay the two columns share the storey shear equally. With those two, the once-indeterminate frame becomes determinate and you can solve it neatly by hand.
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When I raise the column height H, the storey drift jumps up a lot. Is that number really that important?
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Storey drift — the sideways sway — is one of the numbers engineers watch most closely. δ = P·H³/(24EI), so it scales with the cube of the column height H; a tall storey gets very sway-prone fast. Excessive drift cracks the finishes on walls and ceilings, and above all the occupants feel the sway and find it unpleasant. It is a serviceability measure of "how it feels to use", separate from the strength question of whether members break. Raise the stiffness EI, that is, use stockier columns, and the drift drops straight away. Check the chart below.
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Can the portal method always be used?
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It is not a cure-all. The portal method suits the preliminary design of low-rise buildings that are not too tall or slender, where you want a quick estimate of member forces. But because it fixes the inflection points exactly at the mid-points, the error grows when the relative stiffness of columns and beams differs a lot, or for tall, slender buildings. In tall frames the axial deformation of the columns starts to matter, so the cantilever method, which focuses on axial forces, fits better. The final design is verified with the computer stiffness-matrix method. Even so, a rough portal-method hand calculation lets you instantly spot when the computer result is wildly off — a handy sanity check.
Frequently Asked Questions
The portal method is a classic approximate analysis for solving rigid frames (portal frames) under lateral load by hand. A rigid frame is statically indeterminate, so the equations of equilibrium alone cannot find the member forces. The portal method introduces two assumptions: (1) a point of contraflexure (zero bending moment) sits at the mid-height of every column and the mid-span of every beam, and (2) in a single bay, the two columns share the storey shear equally. These two assumptions make the structure determinate, so shears, moments and axial forces follow directly from equilibrium. It is accurate enough for the preliminary design of buildings that are not too tall or slender.
In a single bay the two columns share the storey shear (equal to the lateral load P) equally, so each column shear is V_col = P/2. Because the inflection point sits at the column mid-height, the bending moment at the top and base equals the shear times the half-height: M_col = (P/2)(H/2) = PH/4. The column axial force comes from the overturning moment P*H, which is resisted by a couple formed by the column axial forces over the bay width: N_col = P*H/L, tension in the windward column and compression in the leeward column.
With an inflection point at each column's mid-height, a column behaves as if both ends were rotation-restrained, giving a column stiffness of 12EI/H³. The two columns act in parallel, so the storey stiffness is 24EI/H³. The storey drift (lateral sway) is therefore δ = P·H³/(24EI). Storey drift is a serviceability quantity: excessive drift cracks interior finishes and makes occupants uncomfortable with the sway, so engineers watch it closely alongside member strength.
The portal method suits the preliminary design of low-rise buildings that are not slender, and the sanity-checking of computer analysis, because it gives a quick hand estimate of member forces. Its limit is that it fixes the inflection points exactly at the mid-points; when the relative stiffness of columns and beams differs a lot, or for tall, slender buildings, the error grows. For tall frames where column axial deformation matters, the cantilever method (which focuses on axial forces) is more appropriate. The final design is normally verified with the stiffness (matrix) method via computer analysis.
Real-World Applications
Steel moment frames in buildings: Low- and mid-rise steel office blocks, shops and schools are designed as a continuum of rigid moment frames (portal frames) of columns joined to beams. The chief advantage is that large openings and free floor layouts are possible without diagonal bracing. In the early design stage, engineers use the portal method to quickly estimate the member forces of each storey under wind or earthquake load and set first-cut column and beam sections.
Large open-span industrial buildings: In factories, warehouses and gymnasiums that demand wide column-free space, the portal frame is the skeleton itself. The rigid frame resists the horizontal surge from a travelling crane (crane surge) and the lateral force of strong wind without any bracing. With a long span the beam end moment tends to dominate, so the design of the column-top and beam-end connections becomes critical.
Preliminary seismic and wind design: The portal method gives a quick estimate of how much member force and storey drift each storey and frame line carries under the horizontal force of an earthquake or typhoon. Storey drift in particular is a key serviceability measure (finish cracking, comfort), and the method lets you check early whether the design stays within the allowable drift ratio (roughly 1/200).
Sanity-checking computer analysis: The results of structural-analysis software based on the stiffness-matrix method are cross-checked against a portal-method hand calculation. Confirming that simple values such as the column shear P/2 or the column moment PH/4 are of the right order of magnitude catches load-input mistakes, support-condition errors and modelling errors at an early stage. Approximate solutions still serve in practice as the "check sum" for precise solutions.
Common Misconceptions and Pitfalls
The biggest pitfall is treating the portal-method result as a final design value. The portal method is an approximate analysis that simply assumes the inflection points sit exactly at the mid-points of the columns and beam; it is a tool for preliminary design and for checking. The actual inflection-point positions shift away from the mid-points depending on the relative stiffness of columns and beams, the fixity of the foundation and the conditions of the storeys above and below. For structures where column and beam stiffness differ greatly, or for tall, slender buildings, the error is not negligible. Always verify the final design with computer analysis using the stiffness-matrix method. The values from this tool are for grasping the order of magnitude and rough size.
Next, taking the drift formula δ = PH³/(24EI) at face value. This formula assumes an inflection point at the column mid-height (so the column is close to one with both ends rotation-restrained) and the ideal state in which the beam is stiff enough to almost restrain the rotation of the column tops. A real beam has only finite stiffness, and as the beam deflects the column tops rotate, so the storey drift comes out larger than the calculated value. Also, if the column bases are pinned, the storey stiffness drops from 24EI/H³ to roughly 6EI/H³ and the drift becomes several times larger. Support conditions and beam stiffness have a decisive effect on storey drift.
Finally, forgetting the column axial force and designing the beam and columns for bending alone. In a portal frame the overturning moment P·H is resisted by the couple of tension in the windward column and compression in the leeward column (N_col = PH/L). The smaller the span L, the larger this axial force, and in a tall frame on a narrow span the axial force can govern the column design. If you look only at the bending moment and downplay the axial force, the column in compression risks buckling. Always check the columns for the combined stress of bending plus axial force.
How to Use
Enter portal frame geometry: height (hNum, m) and span (lNum, m). Typical industrial warehouse portals range 4–8 m height and 6–15 m span.
Specify lateral load (pNum, kN) applied at the roof level and column flexural rigidity EI (eiNum, kN·m²). For steel columns, EI ≈ 200 GPa × section moment of inertia.
Click Analyze to compute shear distribution, bending moments, axial forces, and storey drift using the approximate portal method.
Worked Example
Portal frame: height H = 5 m, span L = 10 m, lateral wind load P = 15 kN, column EI = 8500 kN·m² (IPE 360 steel, E = 200 GPa). Portal method splits horizontal shear equally between columns (7.5 kN each). Column base moment ≈ 7.5 × 5 = 37.5 kN·m. Beam midspan moment from frame action ≈ P×H/8 ≈ 9.4 kN·m. Storey drift δ = P×H³/(12×EI_total) ≈ 15×125/(12×17000) ≈ 1.8 mm. Verify against EC3 drift limits (H/300 = 16.7 mm).
Practical Notes
Portal method assumes column shear equals P/2 per column and beam shear equals P. Use for preliminary design; FEA refines results for asymmetric or multi-storey frames.
Column axial forces arise from beam cantilever moments: A_top ≈ (M_beam/L). Check compression + bending interaction (EN 1993-1-1 clause 6.3.4) for I-sections.
Storey drift governs serviceability; typical limits are H/300–H/500 depending on cladding attachment. Increase column EI (use larger sections or bracing) if drift exceeds limits.