Structural Engineering & Statics
where the load goes
Materials

The shear strength nobody measured

Every steel web in a design check is checked against the yield stress divided by the square root of three, and no test produced that number. It is a consequence of a decision about what makes a metal yield, and the alternative decision gives a different answer by fifteen per cent.

Assumes The worst stress is not where the worst bending is, The stress at which nothing in particular happens and Both at once, and neither matters until it does.

18 min read 8 figures Which failure arrives firstGeometry beats materialDrawing as calculation

A steel web is checked against a shear stress of fy/3f_y/\sqrt{3}. For grade 355 steel that is 205 N/mm², it appears in every code, and nobody has ever measured it.

What was measured is fyf_y — a stress in a bar pulled one way. Turning that into a statement about a web, where the stress is shear rather than tension, requires a theory about what makes a metal yield, and the 3\sqrt{3} is the theory rather than the material.

One criterion inside the other, touching at six points. The two yield criteria in principal stress space with the third principal stress zero, both normalised by the yield stress. Von Mises is the ellipse — σ₁² − σ₁σ₂ + σ₂² = f_y², which is a circle seen at an angle — and Tresca is the hexagon inscribed in it, touching at the six points where one principal stress is zero or the two are equal. Everywhere else Tresca is the smaller, by up to 15.5 per cent, and the widest gap is at pure shear, where σ₁ = −σ₂ and the two answers are 205 and 178 N/mm². The ratio there is exactly 2/√3, computed rather than quoted, and it is the whole reason a web is checked against f_y over root three.
Fig. 1 The two criteria in principal stress space with the third principal stress zero. Von Mises is the ellipse — a circle seen at an angle — and Tresca is the hexagon inscribed in it. They touch at six points and disagree everywhere else, by at most 15.5 per cent, and the widest disagreement is at pure shear.

Which free body produced the number

Take an element in pure shear: τ\tau on its faces, nothing else. Its principal stresses are +τ+\tau and −τ-\tau at forty-five degrees to the faces, which is what a Mohr circle says and is the whole of the transformation.

Now ask each criterion when that element yields.

Tresca says a metal yields when the largest shear stress reaches half the uniaxial yield stress — because in the uniaxial test the largest shear is fy/2f_y/2, on the plane at forty-five degrees. In pure shear the largest shear stress is τ\tau itself, so yielding is at τ=fy/2=178\tau = f_y/2 = 178 N/mm².

Von Mises says a metal yields when the distortional part of its strain energy reaches the value it has at uniaxial yield. Written in principal stresses that is σ12−σ1σ2+σ22=fy2\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2 = f_y^2; substituting σ1=−σ2=τ\sigma_1 = -\sigma_2 = \tau gives 3τ2=fy23\tau^2 = f_y^2, so τ=fy/3=205\tau = f_y/\sqrt{3} = 205.

The ratio is exactly 2/3=1.15472/\sqrt{3} = 1.1547, and it is the largest disagreement the two criteria ever have. Everywhere else on the surface they are closer, and at six points they agree exactly — the corners of the hexagon, where one principal stress is zero or the two are equal.

What each of them is actually claiming

The two criteria are not two approximations to a measurement. They are two different physical claims, and both are about what part of a stress state is responsible for yielding.

Tresca’s claim is that yielding is slip on a plane, driven by the shear stress on that plane, so the criterion should involve the largest shear stress and nothing else. That produces a surface made of flat faces — a hexagonal prism in three dimensions — because “the largest of three things” is a piecewise linear function.

Von Mises’s claim is that yielding is driven by the distortion of the element, measured by the strain energy stored in changing its shape rather than its volume. That produces a smooth surface, a circular cylinder, because the distortional energy is a quadratic form.

Both throw away exactly the same thing: the hydrostatic part of the stress. Tresca does it because a pressure adds equally to all three principal stresses and cancels out of every difference; von Mises does it because the distortional energy is defined as what is left after the volumetric part is removed. So both criteria say that squeezing a metal equally in three directions does nothing to it whatever, and both are right about that to remarkable pressures.

One point, every plane through it, one circle. A point carrying 200 N/mm² across one face, 0 across the other and 100 of shear. As the plane is turned, the pair (σ, τ) runs round a circle of radius 141.4 centred at 100.0 — and it goes round at twice the rate the plane does, which is the part always misremembered and the part that makes the picture work. The principal stresses are 241.4 and -41.4, on planes 22.5° from the face the 200 acts on; the largest shear on any plane is 141.4, exactly the radius, and it sits 45° from those — which is 90° round the circle. The von Mises stress that ranks this state against any other is 264.6.
Fig. 2 The transformation the criteria are applied to. A criterion is a statement about principal stresses, and the Mohr circle is what turns a stress state written in the axes a structure was drawn in into the principal stresses a criterion asks for.

Where it turns into a structural check

The criterion becomes a design tool the moment there is more than one stress at a point, which on a beam web is everywhere.

Setting von Mises’s expression for a state of direct stress σ\sigma plus shear τ\tau gives the equivalent stress

σeq=σ2+3τ2\sigma_{eq} = \sqrt{\sigma^2 + 3\tau^2}

and the interaction that follows is an ellipse: (σ/fy)2+(τ/(fy/3))2=1(\sigma/f_y)^2 + (\tau/(f_y/\sqrt3))^2 = 1. Tresca gives σ2+4τ2\sqrt{\sigma^2 + 4\tau^2} and a slightly smaller ellipse.

Bending and shear together, which is where the criterion is actually used. The direct stress a section may carry against the shear stress alongside it, both as fractions of the yield stress. Von Mises gives an ellipse — σ² + 3τ² = f_y² — whose intercept on the shear axis is 0.577f_y, or 205 N/mm² here. Tresca gives σ² + 4τ² = f_y², intercepting at half f_y, 178. The point marked is 200 N/mm² of direct stress with 100 of shear: an equivalent stress of 265 N/mm² and a utilisation of 0.75. A steel web's shear-moment interaction is built on this curve, and the 0.577 in it came from a decision about distortion energy rather than from any test of a web.
Fig. 3 The criterion doing ordinary structural work. A point carrying 200 N/mm² of direct stress and 100 of shear has an equivalent stress of 265 and a utilisation of 0.75 — which no calculation on either stress alone would have produced, since neither is above half the yield stress on its own.

That ellipse is where the moment-shear interaction comes from, one level up: a section carrying both is a collection of points each carrying both, and the section-level interaction is the point-level one integrated. It is why the interaction exists at all, and why it is quadratic in the shear rather than linear.

Which one a code uses, and why it does not much matter

Steel codes use von Mises, and they do so for two reasons that have nothing to do with which is more accurate.

The first is that it is smooth. Tresca’s surface has corners, and a corner is a place where the direction a material flows in is undefined — which is a nuisance for a plasticity calculation and a real difficulty for a finite element program.

The second is that Tresca is the conservative one everywhere, so a code that used it would be leaving up to fifteen per cent on the table with no compensating benefit. Tests on ductile metals sit between the two and closer to von Mises, which settles the question as far as it can be settled.

But the fifteen per cent is not nothing, and it lands in one place. The two criteria agree exactly in uniaxial tension, in uniaxial compression, and in equal biaxial tension — which is to say, in every case a designer meets in a flange. They disagree most in pure shear, which is the case in a web. So the entire practical difference between the two theories is concentrated in the shear check, which is exactly the check that quotes the 3\sqrt{3}.

Nothing happens, and then everything happens. The moment capacity left to a section already carrying shear, against the shear as a fraction of what the web can take. The web holds 29.0% of this section's plastic modulus and the flanges hold the rest, and only the web's share is reduced — by the factor √(1 − v²) that von Mises leaves it. So the curve is flat for most of its length: the first per cent of moment is not lost until v = 0.26, half the shear capacity costs 3.9%, and 16% is not reached until v = 0.9. The tangent at v = 1 is vertical, which is why the last tenth of the shear range costs more than the first eight.
Fig. 4 The section-level version of the same ellipse. A section carrying moment and shear together is not the sum of two independent checks, and the reason is that the material at each point is obeying one criterion rather than two.

The six points where they agree, and why they are the ones everybody tests

The hexagon’s corners are the six stress states in which the two criteria give identical answers, and it is worth naming them because they are almost exactly the set of states anybody has ever tested a metal in.

Two of them are uniaxial tension in each principal direction, and two more are uniaxial compression. The remaining two are equal biaxial tension and equal biaxial compression — states in which two principal stresses are the same, so “the largest difference” and “the distortional energy” happen to scale together.

A tension test cannot distinguish the two theories, and neither can a compression test. The only experiments that can are ones with two unequal non-zero principal stresses: a thin tube under torsion, a tube under combined tension and internal pressure, a cruciform specimen pulled two ways. Those are all difficult, all rare, and all done a century after both criteria were proposed.

That is a general point about theories of this kind rather than a historical remark. A criterion is a rule for extrapolating from the states that are easy to test into the states a structure is actually in, and its whole content is in the region where no data exists. Choosing between two of them is choosing an extrapolation, and the honest form of the choice is the one this essay opened with: fifteen per cent, in the shear check, decided by a preference for smoothness and a small body of thin-tube tests.

The agreement is visible on the interaction curve as well as on the surface, and it is worth drawing at the state a flange is actually in.

Bending and shear together, which is where the criterion is actually used. The direct stress a section may carry against the shear stress alongside it, both as fractions of the yield stress. Von Mises gives an ellipse — σ² + 3τ² = f_y² — whose intercept on the shear axis is 0.577f_y, or 205 N/mm² here. Tresca gives σ² + 4τ² = f_y², intercepting at half f_y, 178. The point marked is 320 N/mm² of direct stress with 20 of shear: an equivalent stress of 322 N/mm² and a utilisation of 0.91. A steel web's shear-moment interaction is built on this curve, and the 0.577 in it came from a decision about distortion energy rather than from any test of a web.
Fig. 5 The third figure’s ellipses again, with the marked point moved from a web to a flange: 320 N/mm² of direct stress alongside 20 of shear. The equivalent stress is 322 and the utilisation 0.91, and the two curves are close enough together at that end of the axis that which criterion is used barely moves the answer. The gap between them opens along the shear axis, where the intercepts are 205 and 178.

Two placements of one picture, then, and the whole practical content of the choice is the distance between the marked points and the shear axis. A flange is checked in the region where the criteria agree; a web is checked in the region where they differ by the full 15.5 per cent. That is also why the transformation matters as much as the criterion — principal stresses are what a criterion consumes, so a stress state written in the axes a structure was drawn in has to be turned into them first, and a section whose principal directions rotate through its depth, as an angle loaded straight down shows, moves along the curve as well as up it.

The materials that do not agree that pressure is irrelevant

The strongest thing about both criteria is the thing that makes them wrong for most of the materials in this collection.

Concrete, rock, soil and dry sand all get stronger under pressure. A confining stress of a twelfth of concrete’s strength raises that strength by half and its ultimate strain by a factor of eight — which is the whole of the confinement argument — and no criterion whose expression contains only stress differences can produce that.

The criteria for those materials therefore have a pressure term. Mohr–Coulomb adds a friction angle and turns the hexagonal prism into a pyramid; Drucker–Prager does the same to the cylinder and gives a cone. The mathematics is the same shape with one extra parameter, and the extra parameter is a friction angle rather than a strength.

A metal does not care what pressure it is under; nothing else agrees. Strength against hydrostatic pressure, for a metal and for a pressure-dependent material. Von Mises's criterion contains only stress differences, so squeezing a metal equally in three directions does nothing at all to it and its locus is a cylinder along the hydrostatic axis — the flat line. A granular material's is a cone: its strength rises with pressure at a rate fixed by its friction angle, 30°, and this is the same statement as the confinement argument, where a lateral pressure of a twelfth of the concrete's strength raises that strength by half. The two are not variants of one theory; they disagree about whether a quantity appears at all.
Fig. 6 Strength against hydrostatic pressure for a metal and for a material with friction in it. The flat line is the whole content of a pressure-independent criterion; the rising one is what a friction angle of 30° does. The two are not variants of a theory — they disagree about whether a quantity appears at all.

And the difference has a design consequence that is easy to state. Confinement is a tool for concrete and not for steel. Wrapping a steel column in a tube does nothing to its material strength; wrapping a concrete one does a great deal, and the reason is one term in an expression.

The deviatoric plane, where the whole surface is one picture

The section drawn at the top of this page is the biaxial one, taken at zero third principal stress. There is a better view of the same object, and it explains the shapes rather than showing them.

Look down the hydrostatic axis — the line σ1=σ2=σ3\sigma_1 = \sigma_2 = \sigma_3 — and project every stress state onto the plane perpendicular to it. That plane is the deviatoric plane, and projecting onto it is exactly the operation of throwing away the pressure that both criteria perform.

Seen that way von Mises is a circle and Tresca is the regular hexagon inscribed in it, touching at six points. Every other section through either surface — including the biaxial ellipse above — is that picture seen at an angle, which is why the ellipse is an ellipse and not something more interesting.

And the ratio 2/32/\sqrt3 is now geometry rather than algebra. It is the ratio of a regular hexagon’s circumradius to its inradius, which is 2/32/\sqrt3 for the same reason an equilateral triangle’s is: the largest disagreement between a circle and an inscribed regular hexagon is at the middle of a side, and pure shear is what the middle of a side corresponds to.

One criterion inside the other, touching at six points. The two yield criteria in principal stress space with the third principal stress zero, both normalised by the yield stress. Von Mises is the ellipse — σ₁² − σ₁σ₂ + σ₂² = f_y², which is a circle seen at an angle — and Tresca is the hexagon inscribed in it, touching at the six points where one principal stress is zero or the two are equal. Everywhere else Tresca is the smaller, by up to 15.5 per cent, and the widest gap is at pure shear, where σ₁ = −σ₂ and the two answers are 159 and 138 N/mm². The ratio there is exactly 2/√3, computed rather than quoted, and it is the whole reason a web is checked against f_y over root three.
Fig. 7 The same pair of criteria at a lower yield stress. Nothing about their relationship changes — the ratio at pure shear is 1.1547 for every metal, because it is a property of the two shapes rather than of either material.

The surface says when, and something else has to say which way

A criterion is only half of a theory of plasticity. It says at what stress states the material begins to flow; it says nothing about the direction the plastic strain goes in when it does, and the second question has consequences the first does not.

The standard answer is the normality rule: the plastic strain increment points along the outward normal to the yield surface at the point the stress state occupies. It is not obvious and it is not arbitrary — it follows from a stability postulate about the material doing non-negative work on itself, and it is what makes the upper- and lower-bound theorems true. Every plastic collapse load on this site rests on it.

Test it against something measurable. For von Mises the surface is a cylinder about the hydrostatic axis, so its normal everywhere lies in the deviatoric plane and has no component along the axis. A strain increment with no hydrostatic component is a volume-preserving one, so normality predicts that plastic flow in a metal changes shape and not volume — which means a plastic Poisson’s ratio of exactly 0.5.

That is checkable in an ordinary tensile test. The elastic Poisson’s ratio of steel is about 0.3; past yield the measured value climbs and approaches 0.5, because the accumulating strain is increasingly plastic and the plastic part contributes nothing to the volume change. A prediction from the shape of a surface, confirmed by a measurement nobody made for that purpose.

Now apply the same rule to a pressure-dependent surface and it goes wrong in an interesting way. Mohr–Coulomb’s surface is a cone, so its normal has a component along the hydrostatic axis — which says that a soil sheared to failure must increase in volume, at a rate set by an angle equal to the friction angle. Dense sand does dilate when it is sheared, which is a striking confirmation. It dilates at a dilation angle of ten or fifteen degrees against a friction angle of thirty or more, which is a flat contradiction of the amount.

So soils and concrete are described by a non-associated flow rule: the surface says when, and a different and flatter surface says which way. That is an untidy theory and it is the honest one, and it costs something specific. The bound theorems require associated flow, so they do not strictly apply to soil at all — the limit analyses that underpin bearing capacity, slope stability and earth pressure are being used outside the conditions of their own proof, on the empirical grounds that the error is small and on the safe side. It is one of the few places in this collection where a standard method is known to rest on an assumption known to be false.

The same rule explains a detail of steel design that otherwise looks like a coincidence. A yielded region in a plate thins as it stretches, because the plastic strain has to be volume-preserving and the in-plane extension has to come from somewhere — which is why a tensile coupon necks, and why material at the root of a notch, restrained from thinning by the elastic material around it, cannot yield at all however high its stress. The triaxial embrittlement of a thick joint is normality applied to a region that is not allowed to obey it.

What a criterion assumes about ductility

There is a quieter assumption underneath all of this, and it is the one that connects this essay to the bound theorems.

A yield criterion is a statement about the surface at which a material starts to flow. Using it as a strength criterion assumes the material goes on flowing at that stress while everything else catches up — that it has a plateau, that the plateau is long enough, and that nothing brittle happens first.

For mild steel that is very nearly exactly true, which is why the whole apparatus works so well there. For a high-strength steel with no yield plateau it is an approximation resting on a proof stress somebody defined. For a bolt, a weld metal or a cast iron it is not true at all, and applying a yield criterion to them predicts the onset of something that will not happen before fracture does.

Put the three curves side by side and the difficulty is visible without any surface being drawn: mild steel has a plateau, and a high-strength steel and an aluminium alloy have a definition. A criterion calibrated on the first and applied to the others is calibrated on a different event, and the number it returns is a stress at which a construction line was cut rather than a stress at which anything was observed to happen.

Where the model stops

Both criteria are isotropic. They assume the material has no preferred direction, which is true of a rolled steel plate to within a few per cent and false of timber, of a fibre composite, and of anything drawn or extruded hard.

Both ignore the Bauschinger effect and the history. A material that has been yielded one way has a lower yield stress the other way, so the surface moves and distorts as it is loaded — which matters for cyclic loading and shakedown and not at all for a monotonic check.

Neither says anything about fracture. A criterion is a statement about the onset of plastic flow, and a material can fail without ever reaching one — by cleavage at low temperature, by fatigue at a fraction of the stress, by a crack that was already there. Those failures belong to a different framework with a different variable.

And the pressure-dependent versions are calibrated rather than derived. A friction angle is a fit to a set of triaxial tests, and the two common surfaces — a pyramid and a cone — are fitted to the same tests in different ways and disagree in the corners in exactly the manner Tresca and von Mises do.

A metal does not care what pressure it is under; nothing else agrees. Strength against hydrostatic pressure, for a metal and for a pressure-dependent material. Von Mises's criterion contains only stress differences, so squeezing a metal equally in three directions does nothing at all to it and its locus is a cylinder along the hydrostatic axis — the flat line. A granular material's is a cone: its strength rises with pressure at a rate fixed by its friction angle, 20°, and this is the same statement as the confinement argument, where a lateral pressure of a twelfth of the concrete's strength raises that strength by half. The two are not variants of one theory; they disagree about whether a quantity appears at all.
Fig. 8 The sixth figure with one number changed: a friction angle of 20° in place of 30°, which is roughly the difference between a stiff clay and a dense sand. The metal’s line does not move, because there is nothing in it for a friction angle to enter; the cone’s slope is the whole of what a triaxial programme is measuring, and it is a fitted parameter rather than a derived one.

The pair makes the asymmetry plain. One of these two loci is a consequence of a claim about distortion and contains no adjustable constant at all; the other is a straight line through a set of test results whose slope is whatever the results gave. A designer using the first is choosing between two theories; a designer using the second is trusting a site investigation.

What the pictures cannot show

Every figure here is a section through a three-dimensional surface, taken at zero third principal stress. That section is the useful one for plates and webs, where one direction is genuinely unstressed, and it is misleading for anything thick: a point inside a heavy weld, a bearing, or a triaxially confined core is somewhere else on the surface entirely, and the plane drawn does not contain it.

Nor can any of them show the property the whole framework rests on, which is that the surface is a surface — that yielding is a well-defined event with a locus. For a material with a smooth transition from elastic to plastic behaviour there is no such locus, only a family of nested surfaces at different amounts of plastic strain, and which one is drawn is a decision.

The assumption the figure rests on

The criterion is applied at a point, and structures are not designed at points.

A web at 265 N/mm² of equivalent stress at one point is not a web that has failed. Steel is ductile: it yields there, sheds stress to its neighbours, and carries more load — which is exactly what the section-level plastic checks in this collection assume and what makes them legitimate. The point-level criterion is the input to a redistribution argument, not a limit on the structure.

Where that stops being safe is where the redistribution has nowhere to go: at a notch, where the surrounding material is elastic and constrains the yielded zone into a triaxial state; at a thick weld, for the same reason; and at low temperature, where the ductility the argument needs is not there. In each of those the yield criterion is still correct about when flow starts and no longer relevant to when the structure fails.

There is a fourth case, and it is the one a code puts a number on. A slender plate buckles locally before it has finished yielding, so how much redistribution a section is entitled to is decided by whether it can reach its own strength rather than by the criterion. The yield criterion says where a point starts to flow; the section classification says whether the section can wait for its neighbours to catch up. Where it cannot, the point-level check is much closer to the whole answer than anybody wants it to be, and the 15.5 per cent this essay is about stops being an accounting detail buried under a redistribution and becomes 15.5 per cent of the capacity.

Where this goes next

One point, every plane through it, one circle. A point carrying 140 N/mm² across one face, 0 across the other and 45 of shear. As the plane is turned, the pair (σ, τ) runs round a circle of radius 83.2 centred at 70.0 — and it goes round at twice the rate the plane does, which is the part always misremembered and the part that makes the picture work. The principal stresses are 153.2 and -13.2, on planes 16.4° from the face the 140 acts on; the largest shear on any plane is 83.2, exactly the radius, and it sits 45° from those — which is 90° round the circle. The von Mises stress that ranks this state against any other is 160.2. The worst stress is not where the worst bending is Every stress this collection has quoted is a stress on a particular plane, and neither the bending stress nor the shear stress is a property of the point. Turn the plane and both change; one pair of numbers does not, and on a short beam it peaks where neither of them does. Nothing happens, and then everything happens. The moment capacity left to a section already carrying shear, against the shear as a fraction of what the web can take. The web holds 26.1% of this section's plastic modulus and the flanges hold the rest, and only the web's share is reduced — by the factor √(1 − v²) that von Mises leaves it. So the curve is flat for most of its length: the first per cent of moment is not lost until v = 0.27, half the shear capacity costs 3.5%, and 15% is not reached until v = 0.9. The tangent at v = 1 is vertical, which is why the last tenth of the shear range costs more than the first eight. Both at once, and neither matters until it does A section carrying shear has less moment capacity, and the reduction is the web's share of the plastic modulus times one minus the root of one minus the shear ratio squared. On a rolled beam that share is a quarter, so half the shear capacity costs three and a half per cent — and then the last tenth costs more than the first eight. The same concrete, held sideways. Two stress-strain curves for one concrete. The lower is a cylinder test: it peaks at 30 N/mm² near a strain of 0.002 and has nothing left by 0.0035, because it fails by splitting apart sideways. The upper is the same material inside a 12 mm hoop at 100 mm centres, which cannot stop it expanding but can make the expansion stretch steel: the lateral pressure of 2.48 N/mm² — 8% of the strength it is multiplying — takes the peak to 44.4 and the ultimate strain to 0.028. The strength gain is 1.48 times and the strain gain 8.0; the area under the curve, which is the toughness, goes up by 11. It is the third number the confinement is provided for. Squeezed sideways into a different material Concrete in a cylinder test fails by splitting apart sideways under a load pushing it down. Put a hoop round it and the splitting has to stretch steel — and a lateral pressure of a twelfth of the strength raises the strength by half and the ultimate strain by eight. Three materials pulled until they stop. Three stress-strain curves — mild steel, high-strength steel, aluminium alloy — plotted to a strain of 2.0%. One of them has a plateau, so the stress at which yielding starts is something the specimen does rather than something anyone chooses. The 0.2% offset construction is drawn on the high-strength steel: a line of slope E from a strain of 0.002, cutting the curve at 460 N/mm². The stress at which nothing in particular happens One material in six has a yield point that a specimen actually does something at. For all the others the yield stress is a construction — a line drawn at an arbitrary offset — and every strength calculation for those materials depends on it.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

The 8 essays that link to this one and share the most of its objects, of 10 that link here.

The objects this essay names

Each one links to every other essay that touches it.

ConfinementDeviatoric stressDuctilityEquivalent stressHydrostatic stressInteraction diagramMohr circlePlasticityPrincipal stressShear yieldTrescaVon misesWeb shearYield criterionYield surface