Summary: Everything published about a weir is written in units that are not angles, and the tightest number on the structure is not a structural one. A thin plate weir exists to produce a discharge figure, and that figure is a tabulated relation between the depth of water over the crest and the volume passing beneath it. Because the crest is both the zero of the head scale and part of the geometry the discharge coefficient is built around, a crest movement enters the answer twice; because discharge goes as the head raised to one and a half or two and a half, the same millimetre of crest movement is worth one and a half, or two and a half times as much flow; and because the condition that most often spoils the answer, submergence, leaves the structure in perfect condition, an instrument watching the concrete is watching one of several ways the number can go wrong. The numbers below convert a crest tilt in degrees into a flow error in percent, in both directions, and are meant to be used as a table.

1. The Number the Weir Exists to Produce

A weir across a channel is one of the oldest flow measuring devices still in daily use, and the reason it has outlasted most of the instruments sent to replace it is that it contains nothing that can drift. There is no rotor, no electronics in the water, and no factory calibration that expires. Its entire output is a published relation between the depth of water over the crest and the volume passing beneath it, and that relation is fixed by the geometry of the opening. A rectangular notch gives one relation and a triangular notch gives another, and both have been measured and tabulated for the better part of a century.

That is also the source of the structure’s particular fragility. A river gauge that drifts can be recalibrated against a reference. A weir that has moved cannot, because the weir is its own calibration. The plate, the notch and the level of the crest are the instrument, and when they change, the discharge figure changes with them without anything appearing to be broken and without anything appearing in an inspection report.

The published uncertainty bands are wider than the reputation of the device suggests, and it is worth stating them before anything is derived from them. International guidance on choosing between gauging structures puts thin plate weirs in a band of 1 to 4 percent, flumes and certain other weirs at 2 to 5 percent, and end methods and the remaining weirs at 4 to 10 percent of the computed discharge. For a 90 degree V notch built and used strictly to the standard, a national measurement programme quotes 1.0 to 1.5 percent. None of those bands is generous, and every one of them assumes that the structure has not moved.

So the question this article answers is narrow and specific. If the concrete moves, by how much is acceptable, and what does that movement cost in the unit the structure exists to produce. It is a question about a conversion, not about structural capacity, and the two answers are not the same number.

2. Where the Crest Enters the Equation Twice

The free discharge relation for a suppressed rectangular sharp crested weir is the width of the notch multiplied by the head raised to the power of one and a half, times a constant that carries the discharge coefficient and the acceleration due to gravity. For a 90 degree V notch the width is not constant, because it grows with depth, so the relation becomes the tangent of half the notch angle multiplied by the head raised to the power of two and a half. In both forms the head is measured from the crest.

The crest is therefore the zero of the head scale, and the governing standard for thin plate weirs, ISO 1438, says so in a procedure rather than in a footnote. The gauge zero is determined by laying a machinist’s level with one end resting on the weir crest and the other end on a temporary hook gauge, reading both, and transferring the difference to the permanent gauge. The crest is not near the datum. The crest is the datum, and the stilling well beside it exists so that the float or the sensor reads a surface that is still, rather than one moving with every wave and every passing boat.

The crest then appears a second time, through the discharge coefficient. The coefficient for a sharp crested weir is not a constant. The relationship published by Rehbock in 1929 makes it depend on the ratio of the head to the height of the crest above the floor of the approach channel, so the same crest level that defines the head also defines part of the coefficient. If the crest settles, the head above it rises and the crest height below it falls, and both of those changes push the coefficient in the same direction.

It follows that a crest movement does not produce a single correction that can be applied to a reading afterwards. It moves the datum and it moves the geometry, and it moves both of them in the direction that makes the structure pass more water than the rating says it is passing. Published guidance on hydraulic structure problems states the consequence plainly for a spillway crest, where uniform settlement lowers the design discharge elevation and results in higher discharges. The same sentence applies to a measuring weir, with one difference: on a weir the consequence is usually a billing question rather than a safety question, which is precisely why it stays unnoticed for longer.

3. Why the Same Millimetre Is Worth One and a Half, or Two and a Half

Differentiating the discharge relation gives the sensitivity directly, and it needs no approximation. The share of error in the discharge is the exponent multiplied by the share of error in the head. On a rectangular notch the exponent is one and a half. On a 90 degree V notch it is two and a half. A national flow measurement programme states the same result the other way round: the percentage error in head measurement is multiplied by approximately 1.5 to give its contribution to the percentage error in flow rate, and that multiplier is why the head reading matters more here than the size of the head would suggest.

Two consequences follow immediately. The first is that a millimetre of head error is not a millimetre of flow error, so a head uncertainty quoted in millimetres cannot be compared with a flow uncertainty quoted in percent without the exponent applied. The second is that the same millimetre is worth 67 percent more on a V notch than on a rectangular notch, from the same instrument, the same installation and the same reading.

That extra sensitivity is not an accident of the shape, it is the reason the shape was chosen. Because the width of a V notch grows with depth, a small flow produces a head large enough to read, and a wide range of flow compresses into a narrow range of head. A 20 to 1 turndown of flow compresses into 3.3 to 1 of head on a 90 degree V notch, against 7.4 to 1 on a rectangular notch. The compression is why a V notch is specified for low flows, and it is also why a fixed millimetre of head error costs more on the notch that was selected for its resolution.

Put a working head on it. At a head of 150 millimetres, one millimetre of head error is 0.67 percent of the head, and therefore 1.0 percent of the flow on a rectangular notch and 1.67 percent of the flow on a V notch. A head device with five millimetres of uncertainty, which is what a good float in a stilling well is worth, is therefore carrying 5.0 percent of flow on the rectangular notch and 8.3 percent on the V notch before any part of the structure has moved at all. Everything that follows is competing with that number.

4. From a Degree to a Flow Rate in Three Steps

An angle measured on a crest is not an answer until it has been carried through three steps. The first is geometry: a crest that tilts by a given angle drops by that angle multiplied by the length of the crest, because a crest behaves as a beam with one end lifted. The second is the head: the drop becomes an error in the depth of water over the crest. The third is the rating: the head error becomes a flow error at the exponent of the weir form.

All three steps are small enough to do in the head, and the reason they are worth writing out is that the first one contains the only number that varies from site to site. A tilt of 0.05 degrees is 0.87 millimetres across a one metre crest and 2.62 millimetres across a three metre crest. The same angle, the same instrument, the same reading, and three times the consequence, because what converts an angle into a length is the length of the thing that rotated.

Two readings of that table are worth separating. The first is the size of the angle that most monitoring programmes would call trivial. Half a degree is a movement a structural inspector would not write down on a form. Across a three metre crest it is 26 millimetres of drop, which at a 150 millimetre working head is a flow error of 27 percent on a rectangular notch and 50 percent on a V notch. The angle that a structural inspection regards as noise is, in the unit the structure exists to produce, larger than the entire uncertainty band the structure was allowed to begin with.

The second reading is the head dependence, which is not intuitive either. The same tilt that is worth 0.87 percent of flow at a 300 millimetre head is worth 2.62 percent at 150 millimetres and 7.85 percent at 50 millimetres on the same rectangular notch, and on a V notch the last of those becomes 13.1 percent. A weir is therefore least accurate at the low flows for which the V notch was explicitly selected, which is the opposite of what the shape of the notch suggests at first glance.

Run the conversion backwards and the tolerance on the structure falls out of the tolerance on the answer, with nothing structural in the calculation at all. If a site is held to 3 percent on flow, and the coefficient and the crest length between them account for 2 percent and half a percent, then the head is allowed 2.18 percent, which at a 150 millimetre head is 2.18 millimetres, which across a three metre crest is 0.042 degrees of tilt. That is the number the crest has to stay inside, and it was derived from a rate of flow, not from a concrete strength.

The first row is the one to read twice. A target of 2 percent on flow cannot be met from the head channel at all, because the coefficient and the crest length have already consumed the budget. The coefficient is not a measurement the site makes; it is a number taken from a table and adopted. When a site sets a flow accuracy target tighter than the accuracy of the coefficient it has chosen, the target is not a requirement, it is a wish, and the honest response is to move the target or to move to a device that is calibrated in place.

5. The Settlement the Structural Engineer Calls Harmless

Settlement of a hydraulic structure is divided in the published literature into two kinds, and the two are treated very differently. Uniform settlement moves the whole structure down without distorting it. Differential settlement moves one part of the structure more than another, and the same literature states plainly that differential settlement is what causes problems for rigid structures, by inducing stresses that lead to joint separation, cracking and shear failure. It adds that differential settlement in a flow surface produces offsets that cause turbulence and can induce cavitation.

For a measuring weir the division runs the other way, and both halves of that statement need to be laid out.

Uniform settlement of a crest moves the datum. The head at every point along the crest grows by the same amount, the crest remains a plane, and the resulting flow error is a constant bias that is indistinguishable from a real change in flow. Nothing on the structure shows it. No crack opens, no joint separates, no concrete spalls, and the record simply becomes wrong by a fixed percentage from the day the settlement happened. It leaves nothing for an inspection to look at, at any stage.

Differential settlement of a crest does not produce a larger flow error than uniform settlement of the same average size, and this is the part that runs against intuition. If the head varies linearly along the crest, the true discharge is the integral of the head raised to the exponent along the length, and the mean head does not give exactly that integral. The difference can be computed rather than argued about, and for a head of 150 millimetres with 2.2 millimetres of average crest drop it comes to 0.0007 percent on a rectangular notch and 0.0033 percent on a V notch. Even at four times that drop it stays under a tenth of a percent. The convexity correction that a careful reader might expect to matter does not, because the exponent is too mild for it to.

The two criteria therefore do not overlap, and that is the practical finding. A structural inspection screens for the differential case, because differential movement is the case that damages the structure. A flow record is spoiled by the uniform case, because uniform movement is the case that leaves no trace on the structure and no entry on an inspection form. A monitoring programme built only on what an inspector would notice is watching the half of the problem that does not affect the number, and the half that affects the number is the half with nothing to look at.

There is one further consequence of the differential case that belongs in the same paragraph rather than in a section of its own. A differentially settled crest becomes an offset in a flow surface, and an offset in a flow surface at high velocity produces turbulence and can lead to cavitation damage on the structure itself. That is a genuine structural and hydraulic failure mode with no analogue in the uniform case, and it is the reason the published literature names differential settlement first even though, for the flow figure, it is not the more damaging of the two.

6. What the Head Device Already Costs

The head is measured by a device, and the published performance of those devices sets the floor under the whole exercise. The international standard for water level measuring devices, ISO 4373, classifies instruments into three performance classes by resolution and nominal uncertainty, and lists typical measurement uncertainties for the types in common field use. A float and counterweight in a stilling well is listed at 5 to 10 millimetres. A radar or laser device, which needs no stilling well and can be mounted above the channel, is listed in the same band. Bubblers, pressure transducers and ultrasonic devices are listed at 10 to 20 millimetres, as is a staff gauge read by a person.

Three findings follow from that table, and the third is the one a monitoring plan is usually built on without noticing.

The first is that the measurement chain is coarser than the movement it is being used to argue about, and in the coarser device types it is much coarser. A crest movement worth discussing is one or two millimetres. The device reading the head is uncertain by 5 to 20 millimetres. Before any structure moves at all, the head channel has already put more uncertainty into the flow number than the settlement mechanism can add, which is why a crest survey carried out with a coarser instrument than the stage device cannot resolve the movement that matters.

The second is that the head chain has its own failure mode, and it is structurally invisible in the same way uniform settlement is. A float in a stilling well drags against its guide, and a published example puts that drag at 0.02 feet of head, which is 6.1 millimetres, on a float six inches in diameter needing two ounces of force to move it. At half a foot of head, that drag alone is a 6 percent flow error on a rectangular notch. A stiff float and a fouled well produce a flow error that no crack gauge and no tiltmeter will ever see, and the remedy is a larger float and less internal friction rather than more instrumentation.

The third is about scope, and it is the reason a crest instrument is worth fitting even though it does not improve the flow figure by itself. A tilt channel on a crest is not part of the flow measurement chain and does not reduce its uncertainty. What it does is audit one of the inputs that the rating assumes, continuously, without a survey crew and without power at the structure. Published guidance names the failure this audit addresses, observing that small systematic errors in head measurement caused by a failure to reference the gauge zero accurately to the crest elevation can be as damaging to accuracy as a drifting instrument, and recommending that the gauge zero be checked whenever necessary. A tilt record is a continuous check on whether the crest and the gauge zero still agree, which is exactly the check that the guidance asks for and that a monthly level run cannot provide during a flood.

7. The Condition Nothing on the Structure Can See

Two families of condition can spoil a rating, and they are different in kind. The first moves the datum, and a tilt channel on the crest is a direct watch on it. The second changes the coefficient, and a tilt channel sees none of it. The largest member of the second family is submergence.

Every free discharge relation assumes that the water below the crest is low enough not to interfere with the nappe. When the tailwater rises to the crest and above it, the weir stops behaving as a weir and starts behaving as a submerged opening, and the free relation overstates the flow. The published modular limits make the tolerance explicit rather than leaving it to judgement: international guidance on gauging structures, ISO 8368, lists a modular limit of about two thirds for a broad crested rectangular profile weir and about three quarters for a triangular profile weir.

Submergence has a published correction. Villemonte proposed it in 1947, and it multiplies the free discharge by a factor built from the ratio of the downstream head to the upstream head, each measured above the crest, raised to the exponent of the weir form and then to the power of 0.385. Computing it shows how much is at stake, and the answer is not marginal.

The numbers in that table are of a different order from the crest movements discussed earlier, and they are produced without any movement at all. A weir sitting at a submergence its own standard permits can be overstating flow by 22 percent while the concrete is in perfect condition, the crest is level, the edge is sharp and the instrument is reading correctly. The correction itself has a validity range and a second published method takes over beyond it, so the correction is a diagnostic rather than a licence; what the table establishes is the order of magnitude, not a working tool for a site that has not measured the tailwater.

The other members of the family change the coefficient rather than the head. A crest that was sharp and has corroded or been fouled to a rounded edge no longer behaves as a thin plate weir; it behaves as a broad crested one, with a different coefficient and a different relation, and the effect on the record is that the reported flow rises while the true flow is unchanged. Because the working record from a weir is a level record, a change in the coefficient is indistinguishable from a change in level. Published guidance is explicit on the mechanism: deviations from the construction, installation or use laid down in the standard produce measurement errors, and a crest edge that has worn over years is a common and silent cause of a works apparently gaining flow.

Put the two families side by side and an honest hierarchy appears. The structural condition is worth single digit percentages. The hydraulic condition is worth tens of percent. A monitoring budget that spends everything on the structure and nothing on the tailwater has bought the smaller half of the risk, and it has bought the half that a tilt channel can already cover.

8. Two Published Numbers That Do Not Agree With Each Other

Two numbers in the published material on weirs are less settled than their presentation suggests, and both matter because both are inputs the site adopts rather than measures. The first is the discharge coefficient.

The relationship published by Rehbock in 1929 is in circulation in at least four forms. The commonest is a constant plus 0.08 times the ratio of the head to the crest height. A hydraulic model published by a national geological survey uses 0.075 in place of 0.08. A federal spillway estimator adds a further term of the head divided by 1000, with the head in feet. A national conservation service directive gives a constant of 0.605 with a reciprocal surface tension term, written as 1 divided by 305 times the head in feet, or 1 divided by 1000 times the head in metres, and the two unit forms of that directive are themselves not numerically equivalent. Evaluated at a head of 150 millimetres and a crest height of 300 millimetres, the forms span about half a percent.

Half a percent is between a twelfth and a half of the 1 to 4 percent band that the whole structure is allowed. That does not make any of the forms wrong, because they were fitted over different ranges and against different sets of data, and the original relationship was itself stated as accurate to half a percent within a defined range of head and crest height. It does mean that the version of the coefficient is a design input with a provenance, and that a site claiming an accuracy tighter than the spread between the versions should be able to say which version it used and over what range it was validated.

The second unsettled number is where the head is measured. Because the water surface draws down as it approaches the crest, the reading has to be taken far enough upstream to be clear of the drawdown and close enough that the energy lost between the gauge and the crest stays negligible. Published sources do not agree on the distance. One edition of the governing standard gives 4 to 5 times the maximum head. A national water meter guide gives 3 to 4. A later edition of the same standard, applied to V notches, gives 2 to 4. A conservation service directive gives 3 to 4 times the maximum expected head.

The spread is 2 to 5 times the maximum head, and the choice is not a detail. The distance is a multiple of the maximum head expected at the site, so a site whose actual maximum head exceeds the one assumed at design has its gauge sitting inside the drawdown, reading a head lower than the true one and therefore understating flow. The unit that has to be written into the record is not the distance, it is the assumed maximum head that the distance was derived from.

Both findings converge on the same practical point, and it is the point that matters for a monitoring plan. On a weir, more of the accuracy comes from the record than from the hardware: which coefficient, over what validated range, with the gauge at what distance upstream, referenced to what crest level, and on what date that reference was last confirmed. An instrument that reports continuously is only useful to the extent that those five entries already exist, because the instrument’s readings are interpreted through them.

9. What a Tilt Channel on a Crest Can and Cannot Certify

A permanently mounted tilt sensor on a weir crest answers a narrow question well, and the question is worth naming precisely. It reports the slope of the crest, continuously, without a survey crew and without power at the structure. Multiplied by the length of the crest, that slope is the drop that the rating is implicitly ignoring, and carried through the exponent of the weir form, that drop is a flow error. The chain is short enough to be run automatically on every reading, which is the whole point: the tilt record can be converted into an allowance against the flow uncertainty the site is held to, and it can be converted every day rather than every time a crew visits.

The accuracy of the instrument is not the limiting term, and this is worth stating with numbers because the intuition usually runs the other way. A tilt sensor with an accuracy of 0.005 degrees, mounted on a one metre crest, resolves 0.09 millimetres of drop. The head measuring device beside it, in the best case the standard lists for a field device, is uncertain by 5 millimetres. The tilt channel is therefore finer than the head channel by a factor of about 60 in the best case the standard lists, and by about 230 against a bubbler or an ultrasonic device. Adding a crest tilt channel does not strain the accuracy budget of a weir; in a typical installation it is the cheapest part of it.

What it cannot certify is larger than what it can, and three limits belong in the same paragraph rather than spread out as qualifications. It cannot see uniform settlement at all, because a crest that drops without rotating produces no slope, and uniform settlement is the case that silently biases the record. It cannot see anything that changes the coefficient, so a crest edge that has rounded, or a plate that has been fouled, is invisible to it. And it cannot see submergence, which is the largest single error in this article and the one the standard’s own modular limit warns about.

The honest scope is therefore a datum audit rather than a flow measurement, and a datum audit is exactly what the published guidance on gauge zero asks for and what a survey crew cannot deliver continuously. What the tilt channel replaces is the periodic confirmation that the crest has not moved relative to the gauge zero. What it does not replace is the upstream stage device, the downstream stage device that would reveal submergence, the inspection that finds a rounded edge, or the calibration of the coefficient itself.

Placement on the structure follows from the criterion rather than from the geometry. A crest is a long, thin element that stands above water for most of its life and is submerged during the events that matter, so the axis to instrument is the one that produces a drop along the crest, and the base over which the drop is computed is the crest length, not the length of the sensor. A reading has no meaning until the length it is divided into has been written down, and if the crest is not straight, the length that matters is the one between the points the rating was referenced to rather than the length of the plate.

One qualification belongs with that, and it applies to V notched weirs rather than to rectangular crests. On a rectangular notch the standard’s requirement is that the crest surface be horizontal and plane, which is a statement about a datum. On a V notch there is no horizontal crest to be level; instead the standard requires the two edges of the notch to subtend equal angles to the horizontal, which turns levelness into a symmetry condition in a rotation that a rectangular notch never has to satisfy. The two rotations are different requirements rather than two views of one, and an instrument reporting a single axis can only be oriented to catch one of them.

The company’s published record in water infrastructure is drainage rather than flow measurement, and it is worth describing precisely so that nothing is implied beyond it. At the Zhanghuabang East drainage system in Baoshan, Shanghai, a rainwater pumping house, a sewage pumping house and a storage tank were combined on a single irregular site roughly 105 metres wide, with the pumping house 10.7 metres deep and the storage tank below it 15.2 metres deep. Twelve inclinometer casings were installed to 30 metres and read daily through the excavation with a two axis probe of 0.005 degrees accuracy and a system repeatability of one millimetre over 30 metres. That is a construction stage installation on a drainage structure, read by hand, and it is not a permanent crest installation on a measuring weir. The relevance is the same conversion, from an angle to a movement that a project was watching, applied to a structure in the same family.

What this article does not claim is that a tilt channel makes a weir accurate. It cannot, because the accuracy of a weir is set by the head device, the coefficient and its version, the gauge distance, the tailwater and the sharpness of the crest. What a tilt channel does is hold one of those inputs under continuous observation, and it is the one that a structural movement can move without leaving any other trace.

Frequently Asked Questions

1. Can a tilt sensor measure the flow over a weir? No, and the two jobs are worth separating. The flow comes from the head, read by a stage device, combined with the published relation for the notch. A tilt sensor measures the slope of the crest, which has to be multiplied by the crest length to become a drop, and carried through the exponent of the weir form to become a flow error. What it produces is an allowance against the accuracy of the rating, not a discharge figure. It checks one of the rating’s inputs rather than replacing the rating.
2. How much crest settlement is too much on a weir? There is no structural answer, because the limit comes from the accuracy the flow figure is required to have. Work backwards: if a site is held to 3 percent on flow, and the coefficient and the crest length account for 2 percent and half a percent, the head is allowed 2.18 millimetres at a 150 millimetre working head, which across a three metre crest is 0.042 degrees. At a 5 percent target the same calculation gives 0.087 degrees, and at 10 percent it gives 0.187 degrees. The tighter figure is set by the abstraction or consent the site operates under, not by a concrete code.
3. Does a level crest guarantee an accurate rating? No. Three conditions sit outside the crest entirely. The nappe has to be fully ventilated, or it clings to the plate and changes the coefficient. The crest edge has to stay sharp, or the structure behaves as a broad crested weir with a different relation. And the flow has to stay modular, meaning the tailwater must stay below the crest; at seven tenths of the upstream head, a published correction puts the true discharge at about 82 percent of what the free relation claims on a V notch and about 71 percent on a rectangular notch. A crest that is perfectly level, with a tailwater that has risen, still produces a wrong number.
4. Is a V notch more sensitive to crest movement than a rectangular notch? Yes, and that sensitivity is the reason the notch was chosen. The exponent is 2.5 against 1.5, so a millimetre of head error is worth 67 percent more flow on a V notch. A V notch is specified for low flows because its width grows with depth, which turns a 20 to 1 range of flow into a 3.3 to 1 range of head; the same property magnifies a fixed head error. A site that selected a V notch for low flow resolution has also selected the form that is hardest on a crest that moves.