Structural Engineering & Statics
where the load goes
Materials

The strength thrown away on purpose

Masonry, concrete and soil are all analysed as though they had no tensile strength whatever. Each of them has some. The decision to set it to zero is the single most consequential modelling assumption in the subject, it is safe for one kind of check and unsafe for another, and almost nothing that uses it says which.

Assumes The middle third, The hinge put in on purpose and The deflection that arrives three years late.

19 min read 10 figures Which failure arrives firstDrawing as calculationThe load must go somewhere

Three of the four materials this collection deals with are analysed as though they could not be pulled at all. Masonry has a tensile strength; so does concrete, at about a tenth of its compressive strength; so, in a small way, does soil. In each case the number is set to zero before any calculation begins, and everything downstream is built on that decision.

It is worth treating as a decision rather than as a fact, because it is one — and because it is safe for one class of question and not for another. Everything below turns on that split: the assumption is protected by a theorem about collapse, and almost nothing a structure has to do every day is collapse.

The middle third, computedThe kern of a 300 × 600 mm rectangle, computed by asking, for every direction, how far the resultant can move before the far face would be pulled. It reaches ±100.0 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±100 of 600 mm33.3% of the depth
Fig. 1 The kern of a rectangle, computed by asking how far the resultant can move in every direction before the far face would be pulled. It reaches h/6 and b/6 exactly, and the region between is a rhombus rather than the ellipse the two numbers suggest.

Which free body produced the number

Cut a section carrying an axial force NN at an eccentricity ee. If the material can be pulled, the stress distribution is N/A±Ne/ZN/A \pm Ne/Z and both faces are in play whatever ee is.

If it cannot, the analysis changes shape entirely. The section is in contact over some part of its width, the stress there is compressive, and the boundary between contact and separation is an unknown to be found rather than a given. The two equilibrium equations then fix two things: the depth of the contact zone and the stress on it.

For a rectangle with a triangular block, the resultant has to act at the centroid of that triangle, so the contact length is 3(t/2−e)3(t/2 - e) and the peak stress follows. That gives the capacity as

N=1.5 f b (t−2e)N = 1.5\,f\,b\,(t - 2e)

— exactly linear in the eccentricity, zero when the resultant reaches the face, and it contains no buckling, no slenderness and no modulus.

A straight line, and the comfortable case is already two thirds down itThe capacity of a 330 mm masonry wall as a fraction of its squash load, against the eccentricity of the resultant. Nothing in this figure is a buckling calculation. A material that cannot be pulled bears on a strip of width 3(t/2 − e) under a triangular stress block, so the capacity is exactly 1.5f(t − 2e) — a straight line, zero when the resultant reaches the face, and already at 67% at the edge of the kern. The middle third is treated everywhere as the comfortable case; a wall loaded there has given away a third of its capacity before slenderness has been mentioned. The lower line is the same wall with slenderness in it, which enters as an ADDITIONAL eccentricity of 16.4 mm rather than as a reduced stress — h_ef²/2400t, for h_ef = 3600 mm. Euler's load for this wall is 13.8 times what the eccentricity rule allows, which is why no masonry calculation contains it.00.10.20.30.40.500.20.40.60.81eccentricity ÷ thicknesscapacity ÷ squash loadthe kern66.7% at the kernno tension only1 − 2e/tslenderness 10.9adds e_a = 16.4 mmEuler is 14×above all of this
Fig. 2 A masonry wall’s capacity against the eccentricity of the resultant. The line is exactly straight, and at the edge of the middle third the capacity is already down to 67% of the squash load.

The middle third, read correctly

The middle third is usually described as the region within which the section stays wholly in compression, which is true, and taken to mean the region within which everything is fine, which is not.

At the edge of the kern the section is on the point of separating, the stress distribution is a triangle with a peak twice the mean, and the capacity is 67% of the squash load. A third of the section’s strength has been given away by an eccentricity that every textbook calls the comfortable limit.

Past the kern it gets worse quickly, because the contact length shortens as the resultant moves and the peak stress therefore rises faster than linearly. Two things are happening at once and only one of them is usually noticed: the section is losing area and the load is becoming more eccentric on the area that is left.

The pressure runs away outside the middle thirdPeak bearing pressure under a 3 × 2 m base carrying 800 kN, against the eccentricity of the load. Inside the middle third the line is straight and the pressure has doubled by the time it reaches the edge of it: 133 kPa at the centre, 267 kPa at e = B/6. Beyond that the base lifts, the contact length shortens, and the curve turns upward without limit — at e = 1.15 m the peak is 770 kPa on 1.04 m of base.00.20.40.60.810200400600800eccentricity of the resultant (m)peak pressure (kPa)B/6: the base is on the point of liftinguniform: 133 kPa
Fig. 3 Peak bearing pressure under a base, against the eccentricity of the load. Inside the middle third the line is straight and the pressure has doubled by the edge of it; beyond that the base lifts and the curve turns upward without limit.

At the centre the pressure is 133 kPa; at the edge of the kern 267; at an eccentricity of 1.15 m on a 3 m base it is 770 kPa on 1.04 m of contact. That runaway is a direct consequence of the no-tension assumption and it is one of the two places where the assumption is not conservative — because a foundation designed on a linear pressure distribution has no idea it is coming.

Why it is safe for strength

The reason the assumption is trusted for capacity is a theorem rather than an argument.

Any stress distribution that satisfies equilibrium with the applied load and nowhere exceeds the material’s strength is a lower bound on the collapse load. That is the static theorem, and two ways of being wrong is the essay about it.

A no-tension analysis produces exactly such a distribution: it is in equilibrium, it uses only compression, and compression is a stress the material certainly can carry. So whatever the material’s real tensile strength is, the answer found without it is safe. The material may be stronger; it cannot be weaker.

A line of thrust, and the masonry it has to stay insideAn arch ring of 8% of the span in thickness, rising 24% of the span, under its own weight as a uniform load. Any horizontal thrust between 4.05 and 5.54 puts a line of compression entirely inside the masonry, so the arch stands — and which of them it actually takes is not decided by statics. The two extremes are drawn: the minimum-thrust line, which rides high at the crown and low at the haunches, and the maximum-thrust line, which does the opposite.thrust anywhere from 4.05 to 5.54 fitsH = 4.05, leastH = 5.54, most
Fig. 4 A line of thrust and the masonry it has to stay inside. Any horizontal thrust between 4.05 and 5.54 puts a line of compression entirely within the ring, so the arch stands — and statics does not decide which of them it takes.

That is the whole of masonry arch analysis. The line that must stay inside is a lower-bound construction: find one thrust line inside the ring and the arch is safe, whatever else it might be doing. It is also why the answer is a range rather than a value — the elastic solution picks one line out of an infinity of admissible ones, and the theorem does not need to know which.

A three-pinned arch, rise 4 on span 16A three-pinned arch under a uniform load. One moment equation about the crown hinge gives a horizontal thrust of 48.00, with no stiffness and no assumption about the section. The thrust line lands on the axis everywhere, so there is no bending anywhere in the arch.crown hinge — no moment here, by constructionH = 48.0H = 48.048.048.0thrust line and axis coincide — the definition of funicular
Fig. 5 The determinate version of the same construction. A three-pinned arch has its thrust settled by one moment equation, with no stiffness and no assumption about the section.

Why it is not safe for stiffness

The other side of the account is the one that gets missed, and it is not protected by any theorem.

The neutral axis is wherever the first moment vanishesA 350 by 556 section with 1400 mm² of steel at a depth of 500, carrying 150 kNm after the tension side has cracked. The neutral axis is no longer at mid-depth: it has risen to 149.7 mm from the top, which is where the first moment of the compression zone plus the transformed steel vanishes. The compression is 12.7 N/mm² at the top fibre and the steel carries 238 N/mm²; the resulting couple is 333 kN on a lever arm of 450 mm, which multiplies back to the 150 kNm applied. The uncracked section would have had 5462×10⁶ mm⁴ against the cracked 1766×10⁶ — a loss of 68% of the stiffness.x = 1501400 mm² of steel, n = 8b = 35012.7 N/mm²333 kN in the steelz = 450C = T = 333 kN · C·z = 150.0 kNm = the applied momentcracked I 1766×10⁶ mm⁴ against uncracked 5462×10⁶ — 68% of the stiffness gone
Fig. 6 A reinforced section after the tension side has cracked. The neutral axis has risen to 150 mm from the top and the second moment has fallen by 68%.

Discarding the tension zone removes most of the section’s second moment. Compute a deflection on the gross section and it is wrong by a factor of three; compute a natural frequency and it is high by the square root of that; compute how a load divides between a cracked member and an uncracked one and the answer is wrong in both directions at once.

None of that is bounded by anything. The lower-bound theorem is about collapse, and it says nothing whatever about a serviceability quantity — so the assumption that makes a strength check safe makes a deflection check unsafe, and the two live in the same calculation.

The awkward middle case is that a member is only as cracked as its moment has made it. A beam is stiffer than its cracked section says because the concrete between the cracks is still working, and where a member sits between the two bounds depends on a tensile strength that has just been set to zero for the other half of the calculation. The same number is being used and not used in one design.

Three materials, three versions of the same assumption

The assumption is made in three places and it does slightly different work in each, which is worth separating because the differences decide where it fails.

Masonry is the pure case. The units have almost no bond to the mortar in tension, the joints open at a stress that cannot be relied on, and the whole of masonry design is a no-tension analysis: the kern, the eccentricity rule, the thrust line, the overturning check. Nothing is added back. A masonry structure’s capacity really is the capacity of a material that cannot be pulled, and the discrepancy between what the calculation gives and what old buildings visibly do is not a modelling error — it is the structure finding a load path that the calculation was not asked about.

Reinforced concrete discards the tension and then puts a different tension path back. The steel is the tension chord, and the calculation is a no-tension analysis of the concrete plus a tension member. So the assumption here is not about capacity at all — it is about where the tension is — and the concrete’s own tensile strength returns as a nuisance rather than a resource: it cracks, it stiffens, it splits covers, and it is the thing anchorage depends on.

Soil discards it because soil genuinely has almost none and because a foundation that lifts is a real and common state. The consequence — a bearing pressure that runs away past the middle third — is the version of the assumption most likely to be met in practice on an ordinary project, since an eccentrically loaded pad is a routine thing to design.

The pattern across the three is that the assumption is least conservative where something is put back in place of the discarded tension, because then the analysis is no longer a pure lower bound on anything.

What else goes with it

Setting the tension to zero discards more than a stress. It discards a mechanism.

The same restraint, twice, with opposite signsA 6 m strip of 200 mm slab whose ends cannot move apart, against deflection measured in its own thicknesses. The flat line is what a yield-line calculation gives, which is what the same strip would carry if its ends were free: 13.3 per unit width. The rising branch is compressive membrane action — the deflected strip is forced into an arch — and it peaks at 42.4, which is 3.18 times the yield-line load, at a deflection of 0.34 of the thickness. Past that the arch runs out of depth and the load falls back to the flexural one; past a deflection of one thickness there is no arch left and the reinforcement starts carrying the strip as a cable. It gets back to the arch's load at 1.64 thicknesses, which is one part in 18 of the span — a sag nobody would design for and exactly what a floor does instead of falling.00.511.522.50102030405060deflection ÷ thicknessload the strip carriesarchingcatenaryyield line× 3.18 at 0.34tnothing to arch against
Fig. 7 A slab strip whose ends cannot move apart. A yield-line calculation gives the flat line; the real strip is forced into an arch and peaks at 3.18 times it.

A restrained slab strip does not bend the way a yield-line analysis says. It is forced into an arch by its own deflection, and the arching carries 3.18 times what the flexural calculation gives — and then, past a deflection of about one thickness, the reinforcement picks the strip up as a cable and it recovers the arch’s load again at a sag of one part in eighteen of the span.

Neither of those two reserves is in any ordinary calculation. They are why a floor slab that has “failed” by every check still holds — the same reserve the force nobody put in the model is about — and they exist because the structure has a tension path even where the material is assumed not to.

Why the number is really discarded

The tensile strength is not thrown away because it is small. A tenth of the compressive strength is not negligible in a section where the tension face is doing most of the work.

It is thrown away because it is unreliable, and in three separate ways.

The same concrete, three sizes, three strengthsThree geometrically similar beams — every dimension in proportion, the same mix, the same notch as a fraction of the depth — failing at nominal stresses of 3.76, 2.97, 1.88 N/mm². The largest is 2.00 times weaker than the smallest, and nothing about the material changed. A strength is being treated as a material property and it is behaving as a property of the specimen, which is what the whole argument is about.d = 25 mm3.76 N/mm²d = 100 mm2.97 N/mm²d = 400 mm1.88 N/mm²every dimension in proportion, including the notchthe largest fails at 50% less nominal stress than the smallest
Fig. 8 Three geometrically similar beams of one mix, failing at nominal stresses of 3.76, 2.97 and 1.88 N/mm². The largest is twice as weak as the smallest and nothing about the material changed.

It has the widest scatter of any property in common use — a coefficient of variation of 20% or more against 5% for a steel yield stress. It is size-dependent, so a value measured on a specimen overstates a member by a factor of two or more. And relying on it produces a brittle failure: a section held together by concrete in tension gives no warning and has no reserve, which is the property ductility supplies and this one removes. That is when half the section has given up read from the other end: a section that has cracked has already discarded its own tension zone, and the calculation is describing what the section did rather than choosing something for it.

Those three together are the argument, and each of them would be enough on its own. What they are not is an argument that the strength is zero, and the difference matters wherever the tensile strength is what stops something — a shrinkage crack, an anchorage splitting, a punching failure.

The crack length at which the strength stops matteringFailure stress against crack length for a toughness of 38 MPa√m, with one steel grade drawn. The falling curve is fracture — Kc divided by Y times the root of pi a — and it does not know what the yield stress is. The horizontal lines are the grades. At 355 N/mm² the two cross at a crack 2.9 mm long. Below that the section yields and the crack is irrelevant; above it the crack decides and the 355 is irrelevant.1020304050600100200300400500crack length, mmstress at failure, N/mm²355 N/mm² crosses at 2.9 mmfracture: the crack decides
Fig. 9 The same unreliability expressed as a length. Below a crack of 2.9 mm the section yields; above it the crack decides and the strength quoted for the material is irrelevant.

The check that changes shape

There is one more consequence of the assumption, and it is structural rather than numerical: it changes what kind of problem a section check is.

With tension allowed, a stress check is a formula. Compute N/A±M/ZN/A \pm M/Z, compare with a limit, done. The section’s properties are known before the load is, the calculation is linear, and superposition holds — two load cases can be added.

With tension forbidden, the contact area depends on the load, so the section properties depend on the answer. The calculation is non-linear, superposition fails, and two load cases cannot be added: the peak pressure under a base carrying dead load plus wind is not the sum of the two peaks.

That is easy to say and easy to violate, because every analysis package computes load combinations by superposing results. A frame model that reports base pressures under each case and adds them has done arithmetic that the material forbids, and the error is in the unsafe direction precisely where the eccentricity is largest.

The same non-linearity is why a no-tension problem sometimes has more than one answer and sometimes none. An arch ring may admit a whole range of thrust lines, as the figure above shows, or none at all — and “none” is the finding that the arch has hinged, which is a collapse rather than a numerical failure.

Where the model stops

A no-tension material still needs a compression check. The lower-bound theorem requires the stress not to exceed the strength anywhere, and a shortening contact zone reaches the compressive limit at some eccentricity. Past the kern, both failures are approaching at once.

The theorem needs ductility. A lower bound is only a bound if the material can redistribute to reach the equilibrium state that was found. Masonry does, by cracking and hinging; a brittle material at a stress concentration does not, and the theorem quietly stops applying.

Slenderness enters as an eccentricity, not as a stress. A masonry wall’s second-order effect is added to ee rather than subtracted from ff, which is why no masonry calculation contains a critical load — Euler’s load for the wall drawn above is 13.8 times what the eccentricity rule allows.

Wrong in shape, right in two integralsThe compression zone of a C45 section with its neutral axis 150 mm down, drawn twice. The curved outline is the real parabolic-rectangular stress distribution — the material's own law read off the linear strain profile plane sections supplies. The rectangle over it is what every design office uses instead: intensity η f_cd = 24.8 MPa over a depth λx = 125 mm. The two shapes are visibly different and give the same answer, because a bending calculation asks a stress distribution only two questions — how much compression there is, and where its resultant acts. Both are 1084 kN at 62.4 mm from the face. The factors are α = 0.8095 and β = 0.4160, and λ = 2β follows from wanting the same centroid. A triangle and a full rectangle match neither integral and are nowhere near.neutral axisC = 1084 kNat βx = 62.4 mmf_cd = 25.5 MPaα = 0.8095 β = 0.4160η f_cd over λxλ = 0.832, η = 0.973same resultant, same position ⇒ same moment, to machine precisiona triangle would be -37% out, a full rectangle 20%
Fig. 10 What the compression side is doing while the tension side is being ignored. Two very different shapes give the same total force at the same position, because a bending calculation asks a stress distribution only two questions.

What the picture cannot show

A no-tension analysis draws a contact zone with a sharp edge. The real section has a zone that is cracked, a zone that is partly cracked and carrying some tension across rough faces, and a zone that is intact — and the boundaries move as the load changes.

Nor does any figure show the assumption being made. It is made in the setting up, before the first line of arithmetic, and it leaves no trace in the output. A stress diagram with nothing below the axis looks exactly like a stress diagram of a material that genuinely cannot be pulled, and a reader cannot tell from it whether the tension was absent or discarded.

The generalisation

The habit worth carrying is to ask, of any simplification, which theorem is protecting it.

The no-tension assumption is protected by the lower-bound theorem, and that theorem is about collapse. So the assumption is safe for collapse and carries no guarantee at all about deflection, cracking, load-sharing, frequency or fatigue — every one of which is decided by a stiffness the assumption has changed.

The same test applies elsewhere. Neglecting a connection’s rotational stiffness is safe for the beam and unsafe for the column. Ignoring composite action is safe for strength and unsafe for the deflection of the floor next to it. Taking a support as pinned is safe for the member and unsafe for whatever the moment went into instead.

In every case the pattern is the same: a simplification made in the direction of less capacity is safe for a capacity check and arbitrary for everything else, because the theorems that make conservatism meaningful are all theorems about collapse, and most of what a structure has to do is not collapse.

Where this goes next

rectangle±100 of 600 mm33.3% of the depth The middle third A material that cannot be pulled imposes a condition on where the load may land, and the condition is a region rather than a point. For a rectangle it is the famous middle third; for every other section it is a shape nobody quotes, and one ordinary section's is nearly twice as generous as the rule allows. thrust anywhere from 3.85 to 5.23 fitsH = 3.85, leastH = 5.23, most The line that must stay inside A masonry arch does not stand because its shape is right. It stands because some line of compression can be drawn inside the stonework — any one will do, and there are infinitely many to choose from. x = 1371200 mm² of steel, n = 7.5b = 30018.0 N/mm²371 kN in the steelz = 404C = T = 371 kN · C·z = 150.0 kNm = the applied momentcracked I 1139×10⁶ mm⁴ against uncracked 3422×10⁶ — 67% of the stiffness gone When half the section has given up Bending theory puts the neutral axis through the centroid. That is a consequence, not a rule — and when the tension side cracks, the same reasoning moves the axis somewhere else entirely. -150-100-505010015000.20.40.60.811.21.4the redundant — horizontal reaction at the right base (kN)load factorbeam 1.00sway 1.25combined 0.71best lower bound 0.714every mechanism is at or above the answer; every admissible field is at or below it Two ways of being wrong Plastic analysis has two theorems and they point in opposite directions. Any equilibrium field that nowhere exceeds the plastic moment gives a load at or below the collapse load; any mechanism gives one at or above it. Only one of those errors is safe.

Named alongside this one

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

The objects this essay names

Each one links to every other essay that touches it.

Bearing pressureBrittlenessCracked sectionEccentricityKernLoad pathLower boundMasonryMembrane actionNo tensionServiceabilitySize effectStress blockTensile strengthThrust line