Section Capacity: Bending, Axial, Shear, and Which One Gives First

Geometry enters differently for each action

A section resists axial load with its AREA and bending with its SECTION MODULUS, and those two scale differently with the same dimensions. Area grows with width times depth. Section modulus, for a rectangle, grows with width times depth SQUARED.

So the two ways of adding material are not equivalent. Doubling a beam's width doubles both its area and its bending capacity. Doubling its DEPTH doubles its area and QUADRUPLES its bending capacity. That is why joists are deep and narrow rather than square, why an I-section puts its material at the extremes and leaves a thin web between, and why the first question about a beam that does not work is whether it can get deeper before it gets wider.

Stiffness is more extreme still: the second moment of area grows with depth CUBED, so doubling depth makes a member eight times stiffer. Bending capacity and deflection therefore respond differently to the same change, which is why a section can pass one check and fail the other.

Reinforced concrete is designed to fail in the steel on purpose

A reinforced concrete section can fail two ways. The steel can yield first, after which the section deflects visibly and cracks widen while it carries on carrying load. Or the concrete can crush first, which happens suddenly and without warning.

Codes deliberately force the first. Limiting the reinforcement ratio — or equivalently requiring a minimum steel strain at failure — guarantees that the steel reaches yield before the concrete reaches its crushing strain. That is what UNDER-REINFORCED means, and it is a safety decision rather than an economy: an over-reinforced section is stronger and is not permitted, because its failure gives no warning.

The same logic runs through the other materials on this list. Ductile steel connections are preferred to brittle bolt failures; masonry shear provisions favour a mode that degrades rather than shatters; timber bending is ductile in compression and brittle in tension, which is why the compression side is allowed to yield. The capacity is only half the design — the MODE is the other half.

Axial and bending interact, and not monotonically

A column carrying both an axial load and a moment is not checked against two separate limits. The two interact, and for a reinforced concrete column that interaction has a shape that surprises people: over the lower part of the range, INCREASING the axial load INCREASES the moment the column can carry.

The reason is that compression closes the tension cracks. Below the balance point, the section is tension-controlled, and axial compression works with the concrete rather than against it. Above the balance point the section is compression-controlled and the usual intuition takes over — more axial load, less moment.

That is why a column is checked against an interaction diagram rather than against a single axial capacity. A calculator that returns a pure axial capacity is giving you one point on that curve — the top of it — and the pages here say so, because using it for a column that also carries moment is reading the wrong point off the right graph.

Shear is a different question, and short deep members fail it first

Bending capacity grows with depth squared; shear capacity grows only with the area of the web. So as a member gets shorter relative to its depth, the shear demand rises against a capacity that has barely moved, and at some span-to-depth ratio shear overtakes bending as the governing action.

That is why deep short beams, transfer beams, corbels and coupling beams have their own design provisions. It is also why a beam designed by bending alone and then shortened is not automatically safe — the check that governed has changed.

NOTCHES are the sharpest case. Cutting a notch in a timber beam does not merely remove its material; it creates a re-entrant corner where shear stress concentrates, and the crack that starts there runs along the grain. The reduction in capacity is far larger than the proportion of section removed, which is why notch depth and position are restricted by rule rather than left to a stress calculation.

Where it fails: sideways, and at the holes

A beam strong in its own plane can fail out of it. LATERAL-TORSIONAL BUCKLING is the mode where the compression flange, unrestrained, buckles sideways and twists the section with it — and the capacity that governs is then a function of the UNBRACED LENGTH rather than of the section alone. The same beam with a floor deck fixed to its top flange and the same beam bare have different capacities, and the difference is large.

This is why the unbraced length appears on the steel pages and why bracing is a design element rather than an afterthought. A beam bracing scheme removed during construction — before the deck is fixed — leaves the member at its weakest at the moment it is being loaded by wet concrete.

Holes are the other case. A castellated or cellular beam has a web with openings, so the shear capacity is the NET section's, and the tee sections above and below an opening also have to carry local bending — an action the plain-section calculation never sees. Web openings cut for services after fabrication are the same problem discovered too late, and they are a common cause of a beam that was adequate becoming inadequate without anyone touching the loads.

The alternatives: full analysis, and the test

The closed-form checks here treat one section under one action set. A real member sits in a frame where the moments depend on the stiffness of everything attached to it, and a full analysis — elastic or, for seismic work, non-linear — redistributes those moments in ways a hand check cannot.

A simply supported beam is the exception, because it is statically determinate: its moments follow from the loads alone. The peak is where the shear crosses zero — wL²/8 at mid-span for a uniform load, Pab/L under a point load, somewhere between when both act — and a point load converts to the uniform load with the same peak moment, 8Pab/L³, which is how a table written in uniform loads can be read for it. That is exact on bending and high on deflection, but low on end shear once the load is nearer a support than a quarter of the span.

For the composite and proprietary cases the honest source is a TEST. A composite column's capacity, a plate connector's, a proprietary hanger's — these come from testing programmes reported in evaluation documents, and the published values already carry the resistance factors that the test programme's scatter justified.

The calculators here size a first pass and check a proportion. They are the right tool for asking whether a section is in the right region before anyone opens an analysis package, and the wrong tool for signing anything off — which every one of them says on the page.