Tapered Beam Design: Shaping Depth to Follow Moment

A beam of constant depth carries the same section along its whole length, even though the bending moment it resists may be large at one point and nearly zero at another. A tapered beam varies its depth so that material is placed where the forces are, and removed where they are not. The result can be lighter, more economical and often more elegant, from steel portal frame rafters to concrete cantilevers and timber roof beams.

This article explains why tapering works, how depth can be matched to the moment diagram, and the engineering effects that a varying section introduces: changes in shear behaviour, more complex deflection calculations, and particular attention to stability.

Why taper a beam?

The bending stress in a beam is the moment divided by the section modulus. In a prismatic beam, the section is sized for the peak moment, so everywhere else the material is under-used. Tapering reduces the section where the moment is low, saving material and self-weight. Self-weight matters most for long spans and cantilevers, where the beam spends much of its capacity carrying itself.

There are other motives too. A tapered beam can provide a roof fall without separate firring, increase headroom near supports, reduce the visual mass of a cantilever, or express the flow of force architecturally. In steel buildings, web-tapered members are fabricated from plate, so the depth can follow the moment diagram closely at little extra cost. In concrete, tapering affects formwork complexity, so it is used where the benefit is clear.

The strongest structural case arises where the moment varies sharply, such as at the knee of a portal frame or the root of a cantilever.

Matching depth to the bending moment

For a rectangular section of constant width b and depth h, the elastic section modulus is Z = b h² / 6 and the second moment of area is I = b h³ / 12. If the design aims for the same bending stress σ everywhere, the depth at any point must satisfy:

h(x) = √(6 M(x) / (b σ))

Consider a cantilever under uniform load w. The moment at distance x from the free end is M = w x² / 2, so the depth for constant stress is proportional to x: a straight linear taper from zero at the tip to a maximum at the root. Under a single point load at the tip, M = P x and the ideal depth varies with the square root of x, a curved profile.

A short example: a cantilever 3 m long, 200 mm wide, carrying 20 kN/m, with an illustrative allowable bending stress of 10 N/mm². The root moment is 20 × 3² / 2 = 90 kNm, so the root depth is √(6 × 90 × 10⁶ / (200 × 10)) ≈ 520 mm. At 1.5 m from the tip, the moment is 22.5 kNm and the required depth is about 260 mm, exactly half, confirming the linear taper. In practice a tapered beam never reduces to zero depth: a minimum depth is kept near the tip for shear, connections and robustness, as shown in our page on cantilever structural design.

Tapered beams in steel portal frames

The most common tapered member in building construction is the haunch of a steel portal frame. Under gravity load, the moment in a portal rafter is greatest at the eaves, where the rafter meets the column. Rather than size the whole rafter for that moment, designers add a haunch: a tapered deepening of the rafter near the eaves, usually cut from the same section as the rafter or fabricated from plate and welded beneath it.

The haunch serves two purposes. It increases the moment capacity where it is needed, allowing a lighter rafter section along the rest of the span, and it provides the lever arm needed for a bolted moment connection at the eaves, as discussed in our page on moment connections. Some portal frames go further, using fully web-tapered rafters and columns fabricated from plate, with depth varying continuously along each member. This is common in pre-engineered metal buildings, where fabrication equipment welds tapered webs to flanges efficiently.

Shear in a tapered beam: the Résal effect

In a prismatic beam, the flanges carry bending forces horizontally and the web carries the vertical shear. In a tapered beam, at least one flange is inclined. The axial force in an inclined flange has a vertical component, and that component shares in carrying the shear. This is known as the Résal effect, named after the French engineer who described it.

The effect can help or harm. Where the beam gets deeper towards the point of maximum moment, as in a cantilever deepening towards its support, the inclined compression flange typically carries part of the shear and relieves the web. In other configurations the flange component adds to the web shear. Ignoring it can therefore be unconservative in some cases and wasteful in others, which is why tapered members need careful shear assessment rather than a prismatic formula applied blindly.

At the point where a flange changes direction, such as the end of a haunch, the change in flange force direction also creates a concentrated transverse force on the web, which may need a stiffener.

Deflection and analysis of tapered beams

Deflection depends on the second moment of area, which in a tapered beam varies continuously. Because I depends on the cube of depth for a rectangle, small changes in depth have large effects on stiffness. Standard deflection formulas for prismatic beams do not apply. Instead, engineers integrate the curvature M/EI along the beam numerically, or divide the member into a series of short prismatic segments in analysis software. Our page on moment of inertia explains why depth dominates stiffness.

A tapered beam optimised for strength can be relatively flexible, since material has been removed from low-moment regions that still contribute to overall stiffness. In long spans and cantilevers where deflection or vibration governs, the taper may need to be less aggressive than strength alone would suggest.

Stability of tapered members

Thin, deep steel sections are vulnerable to lateral torsional buckling, where the compression flange buckles sideways and the section twists. Tapered members are harder to assess because the section properties, and therefore the elastic critical moment, change along the length. Eurocode 3 provides a general method for non-uniform members and the designer may use numerical buckling analysis, with restraint positions and the location of the compression flange carefully considered. In a portal haunch under gravity load, the bottom flange is in compression near the eaves, and restraint is typically provided by fly braces from the purlins or rails.

Local stability matters too. Deep, thin webs can buckle in shear, and slender flanges can buckle locally, so section classification must be checked along the member, not only at one point. Our page on buckling covers the underlying behaviour.

Frequently asked questions

What is a tapered beam?

It is a beam whose depth, and sometimes width, varies along its length. The depth is usually greatest where bending moment is highest, such as at a cantilever root or the eaves of a portal frame, so material is placed where it is needed.

Why are portal frame rafters haunched at the eaves?

The bending moment in the rafter is greatest at the eaves. A haunch deepens the rafter locally, increasing its moment capacity there and providing a deeper lever arm for the bolted eaves connection, so the rest of the rafter can be a lighter section.

Is a tapered beam always more economical?

Not always. It saves material, but fabrication or formwork may cost more, and deflection or stability may limit how much depth can be removed. Tapering is most economical in steel plate fabrication, long spans and cantilevers where self-weight matters.

How is deflection calculated for a tapered beam?

Because stiffness varies along the length, standard prismatic formulas do not apply. Engineers integrate curvature numerically or model the beam as a series of short prismatic segments in analysis software.

Considering a tapered or haunched member for a long span, canopy or cantilever? Talk to our consultancy or get in touch.