Structural Engineering & Statics
where the load goes
Sections and stress

Two strengths, depending which way up

A symmetric section has one section modulus. A tee has two, differing by a factor of three, so the same member has two bending strengths and which applies is decided by the sign of the moment. Turn it over and it is a different beam.

Assumes The material far from the middle does nearly all the work, Bending is a pair of forces, pushing and pulling and Plane sections stay plane, and what the assumption costs.

20 min read 11 figures Geometry beats materialWhich failure arrives first

A symmetric section has one section modulus, and the word “one” in that sentence is doing more work than it looks. Z=I/cZ = I/c needs a distance cc from the neutral axis to the extreme fibre, and a symmetric section has the same distance to both faces, so the ambiguity never arises and the habit of thinking of ZZ as a property of the shape takes hold.

An asymmetric section has two of them, and they can differ by a great deal.

A section modulus for each face, and only the smaller one is a strengthThree profiles of equal area with the second moment divided by BOTH distances to an extreme fibre rather than by the larger of them. A symmetric section has one section modulus and an asymmetric one has two, differing here by as much as 2.92 to one — so the same member has two bending strengths, and which of them applies is decided by the sign of the moment rather than by anything about the section. The bar is the smaller of the two, which is the one that governs when the moment can go either way.tall rectangleZ top 100.0 × 10³Z bottom 100.0 × 10³the same both waysteeZ top 220.1 × 10³Z bottom 75.3 × 10³2.92 : 1I-sectionZ top 242.9 × 10³Z bottom 242.9 × 10³the same both waysthe two solid lines are the extreme fibresthe bar is the smaller section modulus, to scale
Fig. 1 Three profiles of identical area and depth, with the second moment divided by both distances to an extreme fibre rather than by the larger of them. The rectangle and the I-section have one modulus each; the tee has 220.1 × 10³ mm³ to its flange and 75.3 × 10³ to its web tip, a ratio of 2.92 to one.

More second moment, and less strength

Start with the fact that makes the subject worth an essay rather than a footnote.

The tee above has an area of 3,000 mm² and a depth of 200 mm, exactly like the rectangle beside it. Its second moment of area is 11.23×10611.23 \times 10^6 mm⁴ against the rectangle’s 10.00×10610.00 \times 10^6 — twelve per cent more. Its elastic section modulus is 75.3×10375.3 \times 10^3 mm³ against 100.0×103100.0 \times 10^3 — a quarter less.

The same material, three waysThree 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.tall rectangleI = 10.00 × 10⁶1.0× the firstteeI = 11.22 × 10⁶1.1× the firstI-sectionI = 24.29 × 10⁶2.4× the firstevery section here has an area of 3000 — only the shape differsthe bar is the second moment of area, to scale
Fig. 2 The same three profiles read the ordinary way, by second moment of area alone. On this measure the tee beats the rectangle and the two bars are nearly the same length — which is a true statement about deflection and a misleading one about strength.

The two facts are the same fact seen twice. Concentrating material near one face moves the centroid toward it, and moving the centroid toward one face moves it away from the other. Second moment of area rewards the concentration; section modulus divides by the distance to the furthest fibre, and that distance has just grown.

I=11.23×106  ↑ ,cmax⁡=149  mm  ↑ ,Z=Icmax⁡  ↓I = 11.23 \times 10^6 \;\uparrow\,, \qquad c_{\max} = 149\;\text{mm} \;\uparrow\,, \qquad Z = \frac{I}{c_{\max}} \;\downarrow

So the tee is the stiffer member and the weaker one, at the same weight and the same depth. Anything that reads a section catalogue looking for the largest II has just chosen the beam that will yield first.

The factor of forty between four shapes of equal area is a real ranking and it is a ranking on II; the ranking on ZZ is not the same ranking, and for asymmetric shapes it is not even close.

Which free body produced the number

The two moduli are one arithmetic performed twice, and it is worth doing once explicitly.

Cut the beam. On the cut face the direct stress varies linearly with distance from the neutral axis, because plane sections stay plane and nothing else. The resultant of that stress distribution has to be zero force and a moment MM, which is the push and the pull the moment is made of.

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 = 11.22 × 10⁶Z = 75.3 × 10³peak stress 0.0σ = M y ÷ I, at every height
Fig. 3 The stress block on a tee carrying 40 kNm. The zero of the distribution is at the centroid, 149 mm above the web tip, and the two triangles either side of it are unequal in height and in area — the same couple, delivered by two very different distributions.

Zero net force puts the neutral axis at the centroid, which is a statement about the first moment of area, and it puts it at 149 mm from the web tip for this section. The stress at any fibre is then σ=Mc/I\sigma = Mc/I, and the two extreme fibres are at c=149c = 149 and c=51c = 51.

Set the larger of those two stresses to the yield stress and the moment that does it is M=fyI/149=fyZbotM = f_y I / 149 = f_y Z_{bot}. Set the smaller and the moment is fyZtopf_y Z_{top}, which is 2.92 times larger and completely unavailable — the section will have yielded at the other face long before.

Only the smaller modulus is a strength. The larger one is a number about a fibre that never governs, for a moment of that sign.

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. 4 Where the second moment comes from, assembled strip by strip: each strip’s area times the square of its distance from the axis. Nothing in this construction knows where the extreme fibre is, which is precisely why the number it produces is not a strength.
Moving the flanges apartThe second moment of area of an I-section against its depth, with the flange and web areas held constant. The growth is close to quadratic, because the parallel-axis term dominates everything the flanges contribute about their own centres.1001502002503000M10M20M30M40M50M60Moverall depth1.0×2.6×4.9×7.9×13.8×21.3×same steel, moved apart
Fig. 5 And the theorem that assembles it for a built-up shape. The parallel-axis term rewards distance from the centroid, symmetrically — it cannot tell the two sides of the section apart, and neither can any argument that stops at the second moment.

Turn it over and the answer changes

The sign of the moment decides which face is in compression, and for an asymmetric section that decides which modulus governs.

A tee used as a simply supported beam sags: the flange is on top, in compression, and the web tip at the bottom is the extreme tension fibre at c=149c = 149. The governing modulus is 75.3.

The same tee over a support hogs: the flange is now in tension and the web tip is in compression, but the distances have not moved — c=149c = 149 is still the larger. The governing modulus is still 75.3.

So the sign of the moment does not change the elastic strength of a section made of one material. What changes it is turning the section over, which exchanges which physical face is far from the centroid. A tee with its flange at the bottom of a sagging beam has its flange as the extreme tension fibre at c=149c = 149 and its web tip in compression at c=51c = 51 — the same two numbers attached to different pieces of steel.

That distinction is academic in a steel beam, where both faces yield at the same stress, and it is the whole design in three cases where they do not:

Concrete, whose compressive strength is an order of magnitude above its tensile one, so the arrangement that puts the small cc on the tension face is disastrous rather than merely inefficient.

Cast iron, whose tensile strength is about a quarter of its compressive, which is why every cast-iron beam ever made has an enormous bottom flange and a small top one — a tee upside down. The proportions were arrived at empirically before anybody could compute them, and they are the correct answer to this arithmetic.

Any composite arrangement where the two faces are different materials, where the transformed section does the same job with an extra step.

The plastic neutral axis is somewhere else entirely

Push the section past first yield and the neutral axis moves — not gradually toward some limit, but to a completely different geometric definition.

A tee at 60% of its plastic momentThe same tee drawn three ways: the shape, the strain across its depth, and the stress that strain produces in mild steel. The strain diagram is a straight line, because plane sections stay plane whatever the material is doing. The stress diagram is not: 2% of the area has yielded, working inward from both faces, and the neutral axis sits at 149.1 mm against a centroid at 149.0 mm. The compression resultant is 155.7 kN and the tension resultant 155.7 kN, on a lever arm of 142.2 mm, which multiplies back to the 22.2 kNm the section is carrying.neutral axisteestrainalways a straight linestressthe material's own curve, sidewaysC = 155.7 kN · T = 155.7 kN · lever arm 142 mm · M = 22.2 kNm2% of the area has yielded — 0 mm from the top, 10 mm from the bottom · Mp = 36.9 kNm · shape factor 1.78
Fig. 6 The tee part way through, at 60% of its plastic moment: the outer fibres of the web have yielded, the flange is still elastic, and the axis has already begun to move away from the centroid.
A tee at 99% of its plastic momentThe same tee drawn three ways: the shape, the strain across its depth, and the stress that strain produces in mild steel. The strain diagram is a straight line, because plane sections stay plane whatever the material is doing. The stress diagram is not: 58% of the area has yielded, working inward from both faces, and the neutral axis sits at 182.6 mm against a centroid at 149.0 mm. The compression resultant is 343.9 kN and the tension resultant 343.9 kN, on a lever arm of 106.3 mm, which multiplies back to the 36.5 kNm the section is carrying.neutral axisteestrainalways a straight linestressthe material's own curve, sidewaysC = 343.9 kN · T = 343.9 kN · lever arm 106 mm · M = 36.5 kNm58% of the area has yielded — 4 mm from the top, 169 mm from the bottom · Mp = 36.9 kNm · shape factor 1.78
Fig. 7 And near the end of the journey. The stress block is now two rectangles of equal magnitude, so zero net force requires equal areas above and below the axis rather than equal first moments about it.

Elastically, zero net force means the first moment of area about the axis vanishes, which puts it at the centroid. Fully plastic, the stress is ±fy\pm f_y everywhere, so zero net force means the areas above and below are equal. Those are different conditions and they give different answers for any section that is not symmetric.

For this tee: the flange is 1,680 mm² and half the total area is 1,500 mm², so the equal-area axis lies inside the flange, 12.5 mm down from the top — at 187.5 mm from the web tip, against the centroid’s 149.0.

The neutral axis moves 38.5 mm, which is 19% of the section’s depth, entirely because of how hard the section is being pushed. It is not a property of the shape.

The plastic modulus that follows is S=134.3×103S = 134.3 \times 10^3 mm³, and putting it beside the elastic one gives a shape factor of

SZmin⁡=134.375.3=1.78\frac{S}{Z_{\min}} = \frac{134.3}{75.3} = 1.78

against 1.50 for the rectangle and 1.09 for the I-section.

What it costs to reach the plastic moment, for two shapesMoment against curvature for two cross-sections of identical area (3000 mm²) and identical depth (200 mm), in mild steel, each divided by its own first-yield moment and its own first-yield curvature. The tee has a shape factor of 1.78 and reaches 98% of its plastic moment at 8.4 times the curvature at first yield; The rectangle has a shape factor of 1.50 and reaches 98% of its plastic moment at 4.3 times the curvature at first yield. The dashed lines are the rigid-plastic moments, computed from the equal-area axis rather than read off the curves, and no curve reaches its own.02468101200.511.52curvature ÷ curvature at first yieldmoment ÷ moment at first yieldtee: 1.78× the yield moment, at 8.4× the yield curvaturerectangle: 1.50× the yield moment, at 4.3× the yield curvature
Fig. 8 The two sections’ journeys from first yield to full plasticity, computed fibre by fibre. The tee starts lower and finishes closer, because the reserve between the two is exactly what a poor elastic section has that a good one does not.

The bonus is the size of the earlier penalty

A shape factor of 1.78 reads as a merit. It is arithmetically the opposite.

A shape factor is a price quoted as a bonus in general, and for an asymmetric section the quotation is at its most misleading. The tee’s plastic modulus, 134.3, is 89% of the rectangle’s 150.0. Its elastic modulus, 75.3, is 75% of the rectangle’s 100.0. So the tee is 11% weaker than the rectangle plastically and 25% weaker elastically — and the whole of its impressive shape factor is the difference between those two shortfalls.

Plastic capacity very nearly does not care about asymmetry. That is the useful form of the result: at full plasticity the far fibre has no special status, every fibre is at fyf_y, and the only thing that matters is how the area is distributed about the equal-area axis. The elastic calculation is the one that punishes a lopsided shape, and it punishes it for a reason that disappears the moment the material yields.

Which is a genuine argument for plastic design on asymmetric sections and a genuine trap: the reserve is only available if the section can reach it, and a tee’s slender outstanding web in compression frequently cannot.

Where plane sections stop staying planeStrain across a cut face at four span-to-depth ratios, with the straight line the theory assumes drawn faintly behind. For a slender beam the two coincide; for a beam as deep as its span the real distribution is nothing like a straight line, and beam theory has no claim on it.span ÷ depth = 0.02off by 31%span ÷ depth = 0.05off by 31%span ÷ depth = 0.12off by 31%span ÷ depth = 0.25off by 31%the assumption is the theory — everything else is arithmetic on top of it
Fig. 9 The assumption both calculations rest on, and its limits. Plane sections staying plane is what makes the strain distribution linear; neither the elastic nor the plastic argument on this page survives without it, and for a stubby member neither is true.

Add an axial force and both axes move again

Two ways to fail, and the curve between themThe exact plastic interaction between axial force and moment for two sections of identical area, both normalised by their own squash load and their own plastic moment. The tee stands 86.1% of its plastic moment outside the straight line at an axial ratio of 1.00; The rectangle stands 25.0% of its plastic moment outside the straight line at an axial ratio of 0.50. Moments are taken about the equal-area axis, which for the monosymmetric section here is 39 mm from the other one. No curve reaches its own plastic moment, which is what a capacity envelope has to do.The straight line is the rule that says the two capacities share out in proportion, and everything between it and a curve is capacity that rule gives away.00.20.40.60.8100.20.40.60.81moment ÷ plastic momentaxial force ÷ squash loadtee: 86.1% of Mp outside the linerectangle: 25.0% of Mp outside the linethe straight-line rule
Fig. 10 The interaction between axial force and moment for the two shapes. The rectangle’s curve is symmetric about the axis, because the section is; the tee’s is not, and the compression it can carry alongside a sagging moment is not the compression it can carry alongside a hogging one.

An axial force adds a uniform stress to the block, so the elastic zero moves by N/AN/A divided by the stress gradient — and the plastic axis moves so that the area difference above and below carries NN. Both moves happen from different starting points and at different rates.

The consequence for a tee is that its interaction diagram is lopsided, and that a member carrying compression and a moment has four combinations to check rather than two. Nothing about this is exotic: a tee cut from a universal beam is one of the commonest members in a truss, it carries axial force and a moment from the loads applied between its nodes, and both signs of moment occur along its length.

The middle third, computedThe kern of a 300 × 500 mm rectangle, computed by asking, for every direction, how far the resultant can move before the far face would be pulled. It reaches ±83.3 mm vertically and ±50.0 mm horizontally, which are h/6 and b/6 exactly, and the region between is a rhombus rather than the ellipse the two numbers suggest — a resultant on a diagonal has less room than either axis allows.rectangle±83 of 500 mm33.3% of the depth
Fig. 11 And the same asymmetry in the axial problem alone: the kern, the region within which a compressive force can be applied without producing tension anywhere. For a symmetric section it is centred; for a tee it is not, and its two halves are of very different sizes.

Where an asymmetric section is the right answer

Nothing above is an argument against lopsided sections. It is an argument that they are chosen for a reason, and that the reason is never the second moment of area.

Where the two faces are made of different things. A composite floor beam is a steel section with a concrete slab on top of it, and the whole point of the arrangement is that the concrete is in compression and the steel in tension. The transformed section is violently asymmetric — its neutral axis often sits inside the slab — and that is the design working rather than failing.

Where the two faces meet different demands. A crane girder carries a vertical load from the wheel and a horizontal surge from the crab, and the horizontal load acts at the top flange. So a channel is welded along the top of an I-section: the result is asymmetric about the horizontal axis, badly so, and it is the correct member because the two loads arrive at different places.

Where a member is continuous and the moment reverses. A plate girder over several spans has a hogging region and a sagging one, and the flange that governs is different in each. Unequal flanges, or a flange that changes thickness along the span, are the honest response — and they turn a single member into two sections joined by a butt weld, each asymmetric, each right where it is.

And where the material forces it. Cast iron’s tension-to-compression ratio of about one to four asks for a section whose small cc is on the compression face, which is a tee upside down. That the ratio of the two moduli should approximately match the ratio of the two strengths is the design rule, and it says that the right asymmetry is not a compromise but a match — the point at which both faces reach their own limit at the same moment, which is the only arrangement in which no material is wasted.

Where the model stops

Everything here is bending about one axis. A tee has a second axis, about which it is symmetric, and a load applied in neither principal direction gives a beam that moves sideways when it is pushed down. An angle is the section where that matters most and it is asymmetric about both axes.

The plastic argument assumes the whole section can yield. A tee’s web is an outstand supported along one edge only, and if the web tip is in compression it is very likely to buckle locally first — at which point the section cannot reach its own strength and the shape factor is a number about a member that does not exist.

The two moduli are elastic quantities, so they carry the elastic assumptions: no residual stress, no yielding anywhere, and a material with the same modulus in tension and compression. Cast iron fails the last of those, which makes the very case the asymmetry was invented for the case where the arithmetic above is least exact.

And the shear centre has been ignored throughout. A tee’s shear centre is at the junction of the flange and the web, not at the centroid, so a load applied through the centroid twists it — which is a point that is not in the section doing its usual damage to an argument that only looked at bending.

What the pictures cannot show

The stress blocks are drawn at a scale chosen to make both triangles visible. In the elastic figure the two peak stresses differ by the factor 2.92, and drawing them to a common scale makes the small one nearly invisible — which is the honest picture and a poor illustration.

The plastic figures show the neutral axis at its final position, and none of them can show the path: the axis migrates from 149 to 187.5 mm as the moment rises, so the drawing at any single moment ratio is one frame of a movement that has no single answer.

And the second-moment figure assembles II from strips, symmetrically, which is exactly the operation that cannot see the asymmetry this essay is about. It is included because that blindness is the finding.

The ladder from here

Later rungs on this anchor: the angle, asymmetric about both axes, where the principal axes are inclined and nothing decomposes conveniently. The cast-iron beam as a designed object, and the ratios its makers arrived at without the arithmetic. Composite and reinforced sections, where the asymmetry is in the material rather than in the geometry and the transformed section restores it. The tee in a truss, carrying axial force and both signs of moment. The plastic section modulus of a section with a hole in it, where the equal-area axis moves and the elastic one barely does. And the deep question underneath all of it: why the neutral axis is defined by two different conditions at two different load levels, and what that says about treating it as a property of a shape at all.

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. 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. 024681000.511.5curvature ÷ curvature at first yieldmoment ÷ moment at first yieldrectangle: 1.50× the yield moment, at 4.1× the yield curvatureI-section: 1.09× the yield moment, at 1.1× the yield curvature The section that yields from the outside in A rectangle has half again as much moment in reserve past first yield as its elastic capacity suggests, and an I-section has a seventh. Read as a ranking that gets it backwards — the reserve is bought with curvature, and the rectangle pays four times as much of it. neutral axiscompressiontensionI = 29.97 × 10⁶Z = 299.7 × 10³peak stress 200.2σ = M y ÷ I, at every height 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.

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Asymmetric sectionCentroidElastic section modulusEqual area axisExtreme fibreNeutral axisPlastic modulusPlastic neutral axisSecond moment of areaSection modulusShape factorStress block