Structural Engineering & Statics
where the load goes
Sections and stress

Bending is a pair of forces, pushing and pulling

A bending moment is not a mysterious twisting. It is a push near the top of a section and a pull near the bottom, separated by a lever arm — a couple, made out of stress.
11 min read 7 figures Geometry beats materialWhich failure arrives first

A bending moment arrives at a cross-section as a single number, in units of force times length, with no obvious physical picture attached. What the section does with it is much more concrete: it pushes on one side and pulls on the other, and the two forces form a couple that matches the moment.

That is all bending is. The push and the pull are equal, so there is no net force; they are separated, so there is a moment. Everything else — the linear stress distribution, the section modulus, why depth matters — follows from working out how a section arranges that push and pull.

Bending is a push and a pullA section carrying a bending moment, with the stress at every height computed as the moment times the distance from the neutral axis divided by the second moment of area. It is compression above and tension below, and zero exactly at the neutral axis.neutral axiscompressiontensionI = 29.97 × 10⁶Z = 299.7 × 10³peak stress 200.2σ = M y ÷ I, at every height
Fig. 1 A section carrying a bending moment, with the stress at every height computed from the moment, the distance from the neutral axis and the second moment of area. Compression above, tension below, and zero exactly at the neutral axis.

Where the stress comes from

The chain has three links and no gaps.

Geometry. Bending a beam curves it. Fibres on the outside of the curve get longer, fibres on the inside get shorter, and one surface keeps its length. If cross-sections stay flat as the beam bends — the central assumption — then the change in length is proportional to distance from that surface, so the strain is linear in yy.

Material. Elastic material has stress proportional to strain. So the stress is also linear in yy: zero at the neutral axis, greatest at the extreme fibres, opposite in sign above and below.

Equilibrium. The stresses have to add up to the applied moment and to no net force. Zero net force places the neutral axis at the centroid of the area. The moment condition then gives

σ=MyI.\sigma = \frac{My}{I}.

Three sentences, and the only assumption doing real work is the first one.

The couple, and where its arm is

The stress block is triangular on each side, so the resultant of each half acts at the centroid of a triangle — two-thirds of the way out from the neutral axis.

For a rectangle of depth dd, each resultant sits at d/3d/3 from the axis, so the lever arm between push and pull is 2d/32d/3. The compressive resultant is the average stress times the area of the compressed half, which is σmax⁡/2×bd/2\sigma_{\max}/2 \times bd/2.

Multiplying gives M=σmax⁡bd2/6M = \sigma_{\max} bd^2/6, so Z=bd2/6Z = bd^2/6, which is the familiar result obtained without any integration. It is a useful way to hold the section modulus in mind: a fraction of the peak stress, over a fraction of the area, at a fraction of the depth.

The same reasoning explains the I-section immediately. Move nearly all the area to the extreme fibres and two things improve at once — the average stress in the material approaches the peak instead of being half of it, and the lever arm approaches the full depth instead of two-thirds. An ideal I-section with all its area in two thin flanges has Z=Ad/2Z = A d/2, which is fifty percent better than a rectangle of the same area and depth.

Bending is a push and a pullA section carrying a bending moment, with the stress at every height computed as the moment times the distance from the neutral axis divided by the second moment of area. It is compression above and tension below, and zero exactly at the neutral axis.neutral axiscompressiontensionI = 17.33 × 10⁶Z = 173.3 × 10³peak stress 346.2σ = M y ÷ I, at every height
Fig. 2 A solid rectangle carrying the same moment. The stress block is the same shape, but the material near the axis is barely stressed and the peak stress is higher — the section is doing the same job less efficiently.
Bending is a push and a pullA section carrying a bending moment, with the stress at every height computed as the moment times the distance from the neutral axis divided by the second moment of area. It is compression above and tension below, and zero exactly at the neutral axis.neutral axiscompressiontensionI = 21.55 × 10⁶Z = 172.0 × 10³peak stress 348.9σ = M y ÷ I, at every height
Fig. 3 A tee-section. The neutral axis is no longer at mid-depth but at the centroid of the area, so the distance to the extreme fibre differs on the two faces — and the section has two section moduli rather than one.

The tee is the case that catches people. Its stem reaches much further from the neutral axis than its flange does, so the stem’s extreme fibre governs. Turn the load the other way and the flange becomes the tension face, the governing distance changes, and the capacity is quite different. A section with one axis of symmetry has one capacity per direction of bending.

Tension, compression, and which material cares

The two sides of the stress block are not equivalent, because materials are not.

Steel is nearly identical in tension and compression, so a symmetric section makes sense and the extreme fibre on either face governs equally.

Concrete is strong in compression and useless in tension, so the tension side of the block is assumed to have cracked and to carry nothing at all. Reinforcement is placed where the tension would have been, and the couple becomes a concrete compression block against a steel tension force. The lever arm survives; the material on one side of it is replaced.

Timber is stronger in tension than compression along the grain — the opposite of most expectations — so a timber beam usually crushes on its compression face before it tears on its tension face, unless a knot intervenes.

Masonry has no tensile capacity at all, which is a material fact with a geometric consequence, which is why an arch has to keep its thrust line inside the section: the moment a stress block would require tension anywhere, the joint opens instead.

Past the elastic limit

The linear stress block holds only while the material is elastic, and steel is not, past yield.

When the extreme fibre reaches yield, the section has reached its elastic capacity My=σyZM_y = \sigma_y Z. Nothing dramatic happens. The outer fibres stop taking more stress and start taking more strain, while the material further in continues to load up. The stress block spreads from a triangle toward a rectangle.

When the whole section has yielded, the block is fully rectangular and the section can take no more. That moment is the plastic moment, and for a rectangle it is exactly 1.5 times the elastic one — the shape factor. For an I-section, where the material was already near the extremes, the factor is only about 1.14, because there was less under-used material in the middle to recruit.

The consequences are large. A steel beam does not fail when its extreme fibre yields; it forms a plastic hinge at that section and continues to carry load while rotating. In a redundant structure that hinge lets moment redistribute to less-stressed regions, and collapse requires enough hinges to form a mechanism. Plastic design counts hinges rather than stresses, and it exists because the elastic stress block understates what a ductile section can do.

Reading the diagram back to the beam

The stress block is a property of a section; the moment that fills it comes from the whole beam.

Load, shear and moment — a simple spanThe applied load, the shear force it produces and the bending moment that follows, drawn one above another to the same horizontal scale. Shear is the integral of the load and moment is the integral of shear.4 per unit lengthshear16.0moment32.0 at x = 4.00the moment peaks exactly where the shear passes through zero
Fig. 4 The moment diagram along a beam. Each station has its own value, so each station has its own stress block, and the peak of this curve is the section that governs.

Combining the two — the diagram along the beam and the block across the section — gives the design statement in one sentence: the section modulus must be at least the peak moment divided by the permitted stress, everywhere along the beam. Where the moment is small, a smaller section would do — which is what a haunch, a plate curtailment or a tapered rafter is exploiting.

The same material, four waysFour cross-sections of identical area, so identical weight and cost, with the second moment of area computed from each profile's own geometry. Only the arrangement differs, and the stiffest is many times the flattest.the same, laid flatI = 0.06 × 10⁶1.0× the firstsquareI = 0.75 × 10⁶13.3× the firsttall rectangleI = 10.00 × 10⁶177.8× the firstI-sectionI = 24.29 × 10⁶431.8× the firstevery section here has an area of 3000 — only the shape differsthe bar is the second moment of area, to scale
Fig. 5 Four sections of identical area. Each carries the same moment with a different peak stress, because each has a different section modulus — and only one of them is a sensible beam.

Where the stress block comes from

The linear distribution is not an assumption about materials. It is a consequence of an assumption about geometry.

Every strip counts by the square of its distanceA rectangular section divided into equal strips, with each strip's contribution to the second moment of area drawn beside it. The strips are identical in size; only their distance from the neutral axis differs.neutral axiscontribution of each striptotal I = 79.86 × 10⁶the outer strips do almost all of the work
Fig. 6 A section in strips. Plane sections staying plane makes the strain linear in distance, an elastic material makes the stress linear too, and the second moment of area is what results from summing the contributions.

The whole derivation rests on cross-sections remaining flat as the beam bends, which is very accurate for a slender member and progressively wrong as it gets deeper. A beam shorter than about twice its depth does not obey it at all.

The same material, four waysFour cross-sections of identical area, so identical weight and cost, with the second moment of area computed from each profile's own geometry. Only the arrangement differs, and the stiffest is many times the flattest.the same, laid flatI = 0.06 × 10⁶1.0× the firstsquareI = 0.75 × 10⁶13.3× the firsttall rectangleI = 10.00 × 10⁶177.8× the firstI-sectionI = 24.29 × 10⁶431.8× the firstevery section here has an area of 3000 — only the shape differsthe bar is the second moment of area, to scale
Fig. 7 Four sections of identical area carrying the same moment. Each fills its stress block differently, and only one of them uses most of its material near the peak stress.

That is the argument for the I-section stated in terms of the block rather than the integral: an efficient section is one where the average stress is close to the peak, because the material near the neutral axis is being wasted.

Where the model stops

Plane sections stay plane. The assumption behind the linear strain distribution. It fails for deep beams, near concentrated loads, and where shear is large relative to bending.

Linear elastic material. Covered above; the plastic reserve is real and is deliberately excluded from the elastic calculation.

Bending about a principal axis. Load a section about any other axis and it bends in two directions at once. For an angle, whose principal axes are diagonal, the everyday loading direction is not principal, and an angle loaded vertically deflects sideways.

No axial force. A beam-column has a stress block that is the sum of a uniform stress and a bending one, and the neutral axis moves off the centroid — possibly off the section entirely.

No twisting. If the load does not pass through the shear centre, the section twists as well as bends, and a channel loaded through its web is the standard example.

The figures share a specific limitation: the stress block is drawn as a shape beside the section, which suggests the stress lives outside the material. It does not — the block is a graph of stress against height, plotted sideways, and its horizontal axis is a stress rather than a distance. Nothing about the section is that wide.

The ladder from here

Later rungs: plane sections and what the assumption costs. Elastic and plastic section moduli, and shape factors. The plastic hinge and collapse mechanisms. Unsymmetric bending and principal axes. Combined bending and axial force, and the interaction diagram. Composite sections. Cracked reinforced concrete. Prestress, which puts the stress block in backwards before the load arrives. And the shear stress distribution, which peaks exactly where the bending stress does not.

Galileo’s 1638 analysis of a cantilever assumed the whole depth was in tension about the bottom edge, giving a capacity three times too high. The error stood for seventy-five years, and it is the reason the neutral axis is worth naming.

Where this goes next

neutral axiscontribution of each striptotal I = 79.86 × 10⁶the outer strips do almost all of the work The material far from the middle does nearly all the work A strip of steel contributes to bending stiffness in proportion to the square of its distance from the centre. Move the same steel outward and the section gets stiffer for nothing. 2012.57.5shear 7.5moment 22.5the cut, at x = 5nothing was applied here — the internal forces are what the left-hand piece needs What a cut reveals, and why it was there all along Cut a beam anywhere and two quantities appear on the face — a shear force and a bending moment. Nothing was applied there. They are what the material was already doing. the same, laid flatI = 0.06 × 10⁶1.0× the firstsquareI = 0.75 × 10⁶13.3× the firsttall rectangleI = 10.00 × 10⁶177.8× the firstI-sectionI = 24.29 × 10⁶431.8× the firstevery section here has an area of 3000 — only the shape differsthe bar is the second moment of area, to scale The same steel in a different shape, and a factor of forty Four sections of identical area, identical weight and identical cost. The stiffest is dozens of times the stiffest of the flattest, and the only thing that changed was the arrangement.