Beams, columns and frames

Beams and columns are the backbone of most building structures. Beams carry load sideways to their supports through bending; columns carry it down through compression. When beams and columns are joined rigidly they form frames, which can also resist sideways forces. This lesson shows how beams develop shear and bending moment, why deflection matters as much as strength, why columns buckle, and how frames and cantilevers behave. It ends with the practical consequences for construction managers: why you must not notch, core or overload beams, and why temporary column and shore lengths matter.

The formulas here are standard results for simple cases. Real design involves many more checks, done by engineers. Use these results to build intuition and to sanity-check site situations, not to approve structural decisions.

Simply supported beams

A simply supported beam rests on two supports and is free to rotate at each. Under a uniform load w along a span L, each support reaction is half the total load, the shear force is greatest at the supports and falls to zero at mid-span, and the bending moment is zero at the supports and greatest at mid-span.

For a single point load P at mid-span, the reactions are P/2 and the maximum moment is PL/4. These two results, wL²/8 and PL/4, are the most useful formulas in practical structures.

The shape of these diagrams explains where reinforcing and connections are concentrated. In a concrete beam, the bottom bars are heaviest at mid-span where the moment is greatest, and the stirrups are closest together near the supports where the shear is greatest. In a steel beam, holes and copes near supports reduce shear capacity, while flange cuts near mid-span reduce bending capacity.

The span appears squared in the moment formula, so a modest increase in span has a large effect. A beam spanning 10 m carries about 23 percent more moment than one spanning 9 m under the same uniform load. That is why changing a column location on site, or removing a temporary support early, can overload a beam.

Bending stress and section shape

Bending creates tension on one face and compression on the other, with the largest stresses at the extreme fibers. The bending stress depends on the moment and on a property of the cross-section called the section modulus, S. Deep sections have much larger section moduli than shallow ones of the same area, which is why beams are deep and why steel beams are I-shaped: most of the material is in the flanges, far from the center, where it does the most work.

This principle explains several site rules. Drilling a hole near the top or bottom edge of a joist removes material where bending stress is highest, so codes and manufacturers limit holes to the middle third of the depth and away from mid-span. Notching the bottom of a beam at a support reduces shear capacity and can start a split. Engineered joists and I-joists have specific hole charts; follow them exactly.

Deflection

A beam can be strong enough and still be unacceptable if it deflects too much. Excessive deflection cracks finishes and partitions, makes floors bouncy, causes ponding on roofs and jams doors and windows. Codes and design standards limit deflection to a fraction of the span, commonly L/360 for floors under live load and L/240 or L/180 for roofs or total load, depending on what the beam supports.

For a simply supported beam under a uniform load, mid-span deflection depends on the fourth power of the span and inversely on stiffness, the product of the modulus of elasticity E and the moment of inertia I.

Because deflection rises with the fourth power of span, doubling the span multiplies deflection by sixteen. Concrete members also deflect more over time because of creep and shrinkage, which is one reason engineers specify camber, minimum stripping times and long-term deflection checks. Construction managers see this in practice when a slab edge sags and the curtain wall anchors no longer line up, or when a long-span floor needs to be cambered or shored to final levels.

Columns and buckling

A short, stocky column fails by crushing or yielding when the stress reaches the material's strength. A long, slender column fails earlier by buckling: it suddenly bows sideways. The Swiss mathematician Leonhard Euler showed in 1757 that the buckling load of an ideal pin-ended column depends on its stiffness and inversely on the square of its length.

Here K is an effective length factor that depends on how the ends are restrained: about 1.0 for pinned ends, about 0.5 for ends fixed against rotation and 2.0 for a flagpole fixed at the base and free at the top. Because length is squared, doubling the unbraced length of a column cuts its buckling capacity to a quarter. Bracing a column at mid-height halves its effective length and quadruples its buckling capacity.

This is why manufacturers publish shore and scaffold capacities that drop with extension length and that assume specified bracing. Using a shore at a longer extension than its table allows, or leaving out its bracing, can cut its capacity dramatically. Many formwork collapses trace back to exactly this.

Cantilevers

A cantilever is fixed at one end and free at the other, like a balcony, a canopy or an overhanging slab edge. A cantilever has its maximum moment at the support, not in the middle, and the top face is in tension. For a uniform load over a cantilever of length L, the support moment is wL²/2, four times that of a simply supported beam of the same length.

For construction, cantilevers have two important implications. First, their top reinforcing is critical: if top bars are stepped on and pushed down before concrete placement, the cantilever loses much of its strength. Second, cantilevers depend on the back span or the connection to stay in place; removing shores under a cantilever before the back span has gained strength, or before post-tensioning, can lead to failure.

Continuous beams and frames

Most real beams are not simply supported. Concrete beams and slabs are usually continuous over several supports, and steel beams in moment frames are rigidly connected to columns. Continuity reduces mid-span moments and deflections but creates negative moments over the supports, where the top face goes into tension. That is why concrete slabs have top bars over columns and beams, and why stripping or reshoring sequences must respect how the engineer assumed the structure would behave.

A frame is a structure of beams and columns joined so that the joints transfer moment. Moment frames resist lateral loads through the bending of their members. Braced frames resist lateral loads through diagonal braces working in tension and compression. Shear walls resist lateral loads through their in-plane strength. Each lateral system has construction implications covered in later lessons: moment frame connections need careful welding and inspection, braced frames need their braces installed before the frame is loaded laterally, and shear walls need continuous reinforcing and proper construction joints.

What this means on site

The behavior of beams, columns and frames leads directly to site rules that a construction manager should enforce.

• Do not cut, notch, core or drill structural members without the engineer's written approval, and follow manufacturers' hole charts for engineered wood.
• Keep top reinforcing at its specified height, especially in cantilevers and over supports; use chairs and do not walk on bar mats unsupported.
• Do not change spans by moving or removing supports, shores or posts without engineering review.
• Use shores and scaffold frames within their rated extension and with the specified bracing.
• Watch for unexpected deflection, cracking or sagging during construction and stop work to investigate.
• Install permanent bracing and connections in the sequence the erection plan requires.