The section that governs is inside the haunch

The section that changes along the span is about a cantilever, and its finding is a competition: the moment grows linearly from the free end and the section modulus grows as the square of the depth, so the worst section is somewhere along the member and, for a rectangle under a tip load, exactly where the depth has doubled.

A portal frame’s rafter is the same competition with two things changed, and the changes move the answer somewhere a formula cannot go.

The two changes, and what each of them does #

The moment is not a monomial. A cantilever’s is $Px$ or $wx^2/2$ — smooth, monotonic, with a single zero at the free end. A rafter’s is the two end moments the frame delivers plus the free-span parabola of its own load, so it starts at the eaves hogging, falls steeply, crosses zero somewhere in the first third, and rises again to a sagging peak near the apex.

That is already enough to move the check: the utilisation has two local maxima, one in the hogging region and one in the sagging, and which of them governs depends on the frame’s stiffness rather than on the rafter.

And the depth is not a taper — it is a haunch. The section changes over a short length near the corner and is prismatic beyond it, so the modulus has a discontinuity in its slope at the haunch tip. On one side of that station the depth is falling with $x$ and on the other it is constant, while the moment falls smoothly through.

A quantity with a kink in its denominator and none in its numerator has its peak at the kink, unless the numerator gets there first. That is the whole mechanics of this page.

Which free body produced the number #

The free body is the rafter, cut at the eaves and at the apex, with the end moments of the frame applied as they were found.

Crossing the eaves cut are a moment, an axial force and a shear; crossing the apex cut, a moment and an axial force. Along the member is the rafter’s own load. Nothing here is statically determinate: the two end moments come from a frame analysis, and the whole shape of the moment diagram is a property of the frame’s relative stiffnesses rather than of the rafter.

So this calculation sits downstream of the corner moment and cannot be done without it — which is worth noticing, because a haunch changes the stiffness distribution and therefore changes the very end moments it is being checked against. The honest version iterates: haunch, re-analyse, re-check. In practice one pass is enough, because the haunch is short and its effect on the frame’s stiffness is a few per cent.

The axial force is left out here, and that is a real omission rather than a simplification for exposition: a rafter carries a compression of perhaps 100 kN, worth a few per cent of the section’s capacity, and it appears in the real check as an interaction.

Why the tip is the critical station #

Read the two quantities at the two ends of the haunch.

At the eaves the moment is 500 kNm and the section is 906 mm deep, so its plastic modulus is 3.35 × 10⁶ mm³ and its capacity 1,189 kNm. Utilisation 0.42.

At the haunch tip, 3 m along, the moment has fallen to 226 kNm — 45 per cent of its eaves value — and the section has become the bare 453 mm rafter, whose modulus is 1.38 × 10⁶ mm³ and capacity 491 kNm. Utilisation 0.46.

The moment fell by a factor of 2.2 and the capacity by a factor of 2.4. That is the entire result, and it says something general about haunches: since a haunch doubles the depth and the modulus goes as roughly the square of it, the tip governs whenever the moment at the tip is more than about 40 per cent of the moment at the eaves — which for an ordinary haunch length of a tenth of the span it always is.

The consequence for design is a rule that looks arbitrary until this is done. A haunch is checked at its tip, and the eaves is checked for something else — the connection, the column, the web panel in shear. The section with the largest moment on the drawing is not the section that decides the rafter.

Where lengthening it stops helping #

Which raises the obvious question of how long a haunch should be, and the answer has a knee in it.

Past the knee the haunch is protecting a section that is no longer being checked. Every additional millimetre of it is steel, weld and fabrication bought against a limit state that has moved somewhere else — and the somewhere else is the eaves, whose utilisation is fixed by the frame rather than by the haunch.

That is why every rule of thumb about haunch length is a proportion of the span — a tenth, or the distance to the point of contraflexure — rather than a variable to be optimised. The optimum is a knee rather than a minimum, and any length at or a little past the knee is as good as any other.

The profile nobody builds #

The other half of this anchor’s ladder is what happens if the taper is allowed to follow the moment exactly.

The coincidence in that figure is worth a moment. Fully stressed design produces a different curve for every load case: a parabola-rooted profile for a tip load, a straight line for a uniform one, something else for a combination. A member built to any one of them is wrong for the others, and a real structure sees several.

Two load cases, two ideal profiles, and a factor of four between what the straight taper spends against them. That is why fully stressed design is a benchmark rather than a method. It tells how much is available; what it does not know is that a straight cut is cheaper than a curved one, that a rolled section arrives at a fixed depth, and that a member has to be handled, transported and connected. A haunch cut from an offcut of the rafter section — which is how they are actually made — is a straight taper because the offcut is a straight member split diagonally, and the geometry of the fabrication decides the profile before any arithmetic does.

The check that has to be made at a station, not at a section #

There is a habit of mind this page is arguing against and it is worth naming, because it survives long after anybody has stopped believing it.

A member is normally checked at its ends and its middle — three sections, chosen because they are where a prismatic member’s moment diagram has its extremes. That habit is right for a prismatic member and wrong for every member whose capacity varies, and the reason is simple: the quantity being checked is a ratio, and a ratio’s maximum is not at the maximum of its numerator unless the denominator is constant.

Once the denominator varies, the check has to be a sweep. Every member on this site whose section changes has the same requirement: a curtailed bar has to be checked where the steel stops and not where the moment peaks; a plate girder with a flange change has a governing station at each step; a haunched rafter has one at its tip.

The general form is that a step in capacity creates a critical section wherever it lands, and a designer’s job at that point is to place the steps where the demand is falling fastest. That is a different activity from sizing a member, and it is most of what detailing a large girder consists of.

What the tip is worth on a real portal #

Putting the arithmetic on a building makes the size of it clear.

A 30 m span portal at 6 m to the eaves, at 6 m centres, carries perhaps 8 kN/m along each rafter. The frame analysis returns 500 kNm at the eaves and 150 at the apex, and a 457×191×67 rafter has a plastic capacity of 491 kNm.

Without a haunch the rafter fails, at 1.02 of capacity, at the eaves. The next rolled section up — a 533×210×82 — would carry it, at 22 per cent more weight over the whole 15.3 m of every rafter in the building.

With a 3 m haunch it passes at 0.46, and the haunch is 3 m of an offcut of the same section, welded on, at each end of each rafter. That is 20 per cent of a rafter’s length of extra material placed where it does two jobs — the section check here, and the connection’s lever arm — against 22 per cent of a heavier section everywhere.

The comparison is why haunched portals are built the way they are, and why the rafter is nearly always the lightest section that works with a haunch rather than the lightest that works without one. The haunch is not a local repair to a member that is nearly big enough. It is what allows the member to be much smaller than the eaves moment implies, and the tip check is the price of that.

Two haunches, and the one that is not about moment at all #

A portal has a haunch at each eaves and often a small one at the apex, and the two are there for different reasons — which is worth separating because the apex haunch looks like a smaller version of the same thing and is not.

The eaves haunch is what this page has been about: it protects the rafter from the largest moment in the frame and gives the eaves connection its lever arm. Its length is set by the knee in the figure above, its depth by the connection, and its critical station is its tip.

The apex haunch is a connection device and almost nothing else. The moment at the apex is small — 150 kNm against the eaves’ 500 — and the rafter has ample capacity there. What the apex has is a bolted splice between two rafters at an angle, and a splice transmitting a moment needs a depth to do it in. Take the haunch away and the connection needs more bolts than the rafter is deep enough to hold.

So a frame’s two haunches are sized by two different limit states, and only one of them appears on a utilisation curve. That is a common arrangement and it is worth expecting: a member’s geometry is often decided by its connections, and a check that reads only the member reports a comfortable answer about the wrong thing — the same relationship the connection that is not a point describes at every joint on a frame.

Where the model stops #

One load case, and it is not the governing one. A portal is designed for gravity, for wind uplift with the moment diagram reversed, and for a combination — and under uplift the rafter’s sagging region hogs, the haunch is on the wrong side, and the tip check is against a moment of the opposite sign.

Nothing buckles. The haunch’s bottom flange is in compression under gravity load and it is unrestrained — there are no purlins on the inside of a portal — so the real governing check at the haunch is very often lateral-torsional buckling of a tapered member, not the section capacity computed here. That is a stability problem in a member whose section changes, which is one of the genuinely awkward corners of the subject.

The section is checked plastically and the frame elastically. Using a plastic modulus assumes the section can reach it, which needs a class 1 or 2 section and enough rotation capacity — and the haunch tip is exactly where the rotation is demanded if a plastic hinge forms in the rafter.

And the moment diagram is fixed. A haunch stiffens the corner, which attracts moment to it, which raises the eaves moment and lowers the apex one. The check as done here uses the moments from an analysis that may not have contained the haunch.

What the pictures cannot show #

The connection. A haunch exists as much to give the eaves connection its lever arm as to give the rafter its depth: the bolts are in two rows top and bottom of a 900 mm deep end plate rather than a 450 mm one, and the moment they can transmit doubles.

That is the real reason haunches are as deep as they are, and no utilisation curve contains it. A haunch sized purely by the rafter check would be shallower and longer than the ones that get built; sized by the connection it is deeper and shorter, and then the tip check has to be done on whatever geometry results.

They also cannot show the fabrication. The haunch in the figures is a taper from 906 to 453 — exactly double — because it is cut from a length of the same rolled section, split along a diagonal and welded to the underside. That construction fixes the depth ratio at two and makes the taper straight, and it is the reason no portal frame in the country has a haunch of 1.7 times the rafter depth.

The assumption the figure rests on #

That the section modulus at a station describes the capacity there.

In a haunch it very nearly does not. The extra depth comes from a cut length of section welded on, so the haunch’s bottom flange is a separate member stopping abruptly at the tip, and the stress it carries has to be transferred into the rafter’s own flange over some length rather than at a point.

So the real distribution of capacity has a transition zone at the tip, a few hundred millimetres long, in which the section is neither the haunch nor the rafter and the flange force is being handed over through the web. That region is where haunch failures are observed, it is why the tip stiffener exists, and it is invisible to any calculation that treats the depth as a function and the section as complete at every station — the same Saint-Venant transition that appears wherever a section changes abruptly.

What to carry away #

Three things, and the third is the one that transfers.

A haunched member’s critical section is at the haunch tip whenever the moment there is more than about 40 per cent of the moment at the eaves, which for ordinary haunch lengths it always is. The check is a sweep along the member rather than an inspection of its ends.

Lengthening a haunch buys capacity steeply until the tip reaches a low-moment region and then buys nothing at all. The optimum is a knee, so any length near it is as good as any other, which is why the rules of thumb work.

And a member whose capacity varies has to be checked as a ratio along its length. The moment diagram is only half of what decides a critical section; the other half is what the fabricator did to the section, and the two have to be divided into one another before anything is known.

The ladder from here #

Later rungs on this anchor: lateral-torsional buckling of a tapered member, where the varying section changes the critical moment and no equivalent uniform moment factor quite works. The stepped member — a plate girder with flange curtailment — which is this problem with a discontinuous capacity and a governing section at every step. The haunch’s own transition region, with the flange force transfer and the stiffener that carries it. Tapered members in plastic design, where the hinge has to form at a section that is not the deepest one. And the fabrication question in full: what a taper costs in cutting, welding and handling against what it saves in steel, which is the calculation that decides whether any of this is worth doing.