Beams

What it does

Carry transverse loads across an opening without excessive stress or deflection.

Change a beam's depth, span and loading and follow bending and deflection together.

When it fits

Learning the effect of geometry on a simply supported timber beam.

The trade-offs

The reference strength line omits the adjustment factors and other checks required for a design.

Context. An educational model. Geometry, material assumptions and the reference building are described in the illustrated study.

A beam takes loads that push across it and carries them sideways to its supports. It can only do that by bending, and every interesting thing about beams follows from that bend: where the material is squeezed and where it is stretched, how far the beam sags, and why a beam turned on its edge is so much stronger than the same piece of timber laid flat. The bench above lets you load one and watch.

Bending

Load the beam and it curves. The fibres along the top get shorter and the fibres along the bottom get longer: the top is in compression, the bottom in tension. Somewhere between them is a layer that does neither, the neutral axis, dashed on the beam’s face. In a rectangular section it runs through the middle.

The grid drawn on the beam shows the rule that makes beams calculable: plane sections stay plane. Each vertical line stays straight as the beam bends; it only tips. So the strain in a fibre grows in straight proportion to its distance from the neutral axis, and so does the stress. The extreme fibres, top and bottom, work hardest.

At any cut through the beam, the loads on one side try to turn it about the cut. That turning effect is the bending moment. The beam answers with an equal resisting moment: the compression above the neutral axis and the tension below it form a couple. The moment diagram standing under the bench draws the bending moment at every point of the span. Where it peaks, the fibres are stressed most.

σ = M / Sbending stress = bending moment ÷ section modulus

Deflection

How far the beam sags depends on four things: the load, the span, the stiffness of the material (its modulus of elasticity, E) and the shape of the section (its moment of inertia, I). For a load spread evenly along a simple beam:

δ = 5 w L⁴ / 384 E Ithe span enters to the fourth power

Lengthen the span and watch the gauge: doubling the span with the same load per foot makes the beam sag sixteen times as much. Building codes cap deflection as a fraction of the span: commonly L/360 under live load for floors with plaster or brittle finishes, and L/240 under the total load. A floor can be perfectly safe and still feel springy or crack a ceiling; deflection, not strength, often decides the size of a long beam. The drawing exaggerates the sag about six times so you can see it; the readout gives the true value.

Depth is everything

For a rectangle of width b and depth d, the two section properties are:

I = b d³ / 12 S = b d² / 6

Width counts once, depth counts twice for strength and three times for stiffness. Double the width and the beam is twice as strong and twice as stiff. Double the depth and it is four times as strong and eight times as stiff. Switch the depth from 9 to 18 inches and the dial barely moves. This is why joists stand on edge, why glulam beams are tall and narrow, and why a steel I-beam puts nearly all its metal in the flanges, as far from the neutral axis as it can.

Change	Bending stress	Deflection
Double the load	× 2	× 2
Double the width	× ½	× ½
Double the depth	× ¼	× ⅛
Double the span, same load per foot	× 4	× 16
Double the span, same total load	× 2	× 8

Shear

Bending is not the only thing going on. At any cut, the loads on one side also try to slide that part of the beam past the other: transverse shear. It is largest near the supports, where the whole reaction passes through. Inside the beam it shows up as horizontal shear too, the tendency of the layers to slip over one another, like the pages of a book when you bend it. In a rectangular section that stress peaks at the neutral axis at 1.5 times the average, which is why the glue lines of a laminated beam matter as much as the wood.

Supports and equilibrium

A body at rest has no net force and no net turning on it. For the beam, the upward reactions at the supports must add up to the load, and their moments about any point must cancel the load’s. The load cells under the bearings show this: whatever you stack on the beam, their two readings always add up to the weight of the blocks plus the beam. Slide a single stack toward one end and that end’s cell takes more:

R_A = P b / L R_B = P a / La and b are the distances from the load to each support

One end sits on a pin and the other on rollers, so the beam is free to rotate at its ends and to lengthen. That makes it a simple beam, and statically determinate: its reactions, shears and moments come from equilibrium alone, whatever the beam is made of.

Other ways to hold a beam

How the ends are held changes everything. Each common case has its own peak moment and deflection, here for a uniform load w over a span L:

Beam	Peak moment	Peak deflection
Simple	wL² / 8 at midspan	5wL⁴ / 384EI
Simple, point load P at midspan	PL / 4	PL³ / 48EI
Cantilever	wL² / 2 at the support	wL⁴ / 8EI at the tip
Fixed at both ends	wL² / 12 at the supports	wL⁴ / 384EI

An overhanging beam runs past a support. The overhang bends the other way and reduces the moment in the span; with one overhang of about three-eighths of the span, the moment over the support equals the moment at midspan. Continuous beams run over several supports and are stiffer than a row of simple beams, but they are indeterminate: their moments depend on the stiffness of each span, not on equilibrium alone.

In the house

The beam on the bench is the reference house’s girder: a 5⅛ × 12 in glulam down the middle of the basement, carrying the inner ends of the first-floor joists and the bearing wall above them, on two steel columns 12 ft apart. Every house with a floor wider than its joists can span has one, and there are several ways to make it.

What carries the floor down the middle?

Decided by the span from wall to wall · how many posts the basement can have · headroom under it · what the crew can lift

Built-up lumber
Three or four 2×10s or 2×12s nailed side by side, their joints staggered over posts.
Choose it when posts can stand every 7 to 9 ft; the crew wants to carry and nail it in pieces; the cheapest girder will do.
The catch many posts in the basement; it shrinks like any sawn lumber; joints between plies must land on posts.
GlulamThis house
Kiln-dried laminations glued under pressure into one deep, straight beam.
Choose it when spans of 12 to 20 ft between posts; the beam will be seen; it must stay straight and barely shrink.
The catch made to order in set widths; heavier to place than built-up plies.
The girder in the house No. 03 →
LVL
Laminated veneer lumber: thin veneers glued with all their grain one way, sold in plies 1¾ in thick.
Choose it when a strong, stiff beam is needed in the same depth as the joists; the plies must be carried in by hand and nailed together in place.
The catch costlier than lumber; must be kept dry on site; plies nailed or bolted to the maker's schedule.
Steel W-beam
A wide-flange rolled steel beam, often with a wood plate bolted on top for the joists to nail to.
Choose it when the longest spans or the fewest posts; the least depth for the load, for headroom.
The catch a crane or a crew to set it; a different trade to fabricate it; a cold line to insulate where it meets the outside wall.
Bearing wall
A stud wall on a strip footing in the basement, carrying the joists along its top plate.
Choose it when the basement will be divided anyway; no beam is wanted at all.
The catch a wall where an open basement might have been; a footing along its whole length.

Here A 5⅛ × 12 in glulam spans 12 ft between two steel columns, so the 36 ft basement has just two posts, and its top is flush with the sill so the joists bear on it level.

The numbers in the drawing

Beam	24F-V4 Douglas fir glulam, 5⅛ in (130) wide, 1½ in (38) laminations
Stiffness	E = 1.8 × 10⁶ psi (12.4 GPa)
Capacity line	Fb = 2,400 psi (16.5 MPa), the reference value before adjustment factors
Self weight	Density 530 kg/m³, included in every result
Blocks	Cast iron, 50 kg (110 lb) each
Method	Exact superposition of each stack as a point load, plus the beam’s weight

The capacity line is a teaching reference, not a design check. A real design adjusts the allowable stress for how long the load lasts, the moisture in service, the size of the beam and how well it is braced against twisting sideways. For timber beams in buildings, see Floor systems.