Summary: The conversion itself is one multiplication: horizontal displacement equals span multiplied by the tangent of the angle, and one degree equals 17.46 millimetres per metre. The difficulty is not the arithmetic but the error budget. On a high-accuracy tilt sensor the zero deviation and the zero temperature drift are an order of magnitude larger than the stated accuracy, so over a ten metre span the accuracy term contributes under one millimetre while the drift term can contribute more than twenty. Treat those terms as stable offsets rather than noise: reading against a baseline cancels them, segmental summation rebuilds the deflection profile, and a single sensor can never separate rigid rotation from bending.

 

1. Why Monitoring Plans Are Written in Millimetres and Instruments Read in Degrees

Almost every acceptance criterion in structural and geotechnical monitoring is expressed either as a length or as a ratio of two lengths. Angular distortion is defined as a differential movement divided by the distance between two reference points. Deflection limits for bridge girders are written as the span divided by 400, 500 or 600. Dam and pier criteria are quoted in millimetres of horizontal movement. These thresholds are expressed that way because they descend from the damage mechanism itself: cracking in a masonry wall begins when the tensile strain in the wall reaches a critical value, and tensile strain is a length change per unit length.

The instruments installed on those same structures, however, report degrees. A MEMS tilt sensor measures the projection of gravity on its sensitive axis and converts it to an angle. So every tilt-based monitoring plan contains a conversion step between what the sensor publishes and what the specification demands, and that step is frequently left implicit. When it stays implicit, two things go wrong: nobody checks which error term dominates once the angle is multiplied by a span, and nobody notices that the largest terms in the datasheet are not the ones being compared in the procurement table.

This guide works the conversion in the open, in both directions, and then examines what the datasheet numbers actually become once they are expressed in millimetres.

 

2. The Conversion: One Degree Is 17.46 Millimetres per Metre

For a structure segment of length L that rotates through an angle theta, the horizontal movement of one end relative to the other is the tangent of that angle multiplied by the length:

Displacement = L x tan(theta) where L is the span or segment length in metres and theta is the rotation in degrees.

For the angles encountered in monitoring, which are almost always well below one degree, the three small-angle forms are interchangeable. The tangent, the sine and the angle in radians agree to within a few parts per million at 0.2 degrees, far below any other uncertainty in the chain, so an instrument manual that publishes a sine formula and an analysis paper that publishes a tangent formula are describing the same relationship. The choice of form changes nothing measurable.

What is worth memorising is the constant. One degree of rotation produces 17.46 millimetres of movement per metre of span, so a thousandth of a degree produces 0.0175 millimetres per metre. That single number converts any angular specification into a length without a calculator, and it runs in both directions: if a specification demands that you resolve 2 millimetres of movement over a 10 metre span, you need to resolve 0.0115 degrees.

Angle change	Movement per metre (mm/m)	Over a 10 m span (mm)	Over a 20 m span (mm)
0.001 degrees	0.0175	0.17	0.35
0.005 degrees	0.087	0.87	1.75
0.01 degrees	0.175	1.75	3.49
0.05 degrees	0.873	8.73	17.46
0.1 degrees	1.75	17.46	34.91
1 degree	17.46	174.6	349.1

The published damage thresholds are small in angle terms. The classical settlement damage criteria of Skempton and MacDonald relate angular distortion to cracking, and their values read as ratios rather than degrees: 1/500 is the recommended limit to avoid cracking, 1/300 corresponds to first cracking in panel and load bearing walls, and 1/150 corresponds to severe cracking and structural damage. Converted, those become 0.115, 0.191 and 0.382 degrees. Over a 12 metre building length, the first cracking threshold is only 40 millimetres of differential movement, and the no-cracking limit is 24 millimetres.

Angular distortion	Equivalent rotation	Differential movement over 12 m	Significance
1/1000	0.057 degrees	12 mm	Comfortably below the cracking threshold
1/500	0.115 degrees	24 mm	Recommended limit to avoid cracking
1/300	0.191 degrees	40 mm	First cracking in panels and load bearing walls
1/150	0.382 degrees	80 mm	Severe cracking and structural damage

Two conclusions follow immediately. First, the rotations that matter are tiny, which is why precision tilt sensors are specified in thousandths of a degree. Second, the span is a multiplier, which is what makes the error budget in section 4 the real subject of this article.

 

3. Why One Sensor Cannot Give You a Deflection

A single tilt reading describes the orientation of a surface. It does not describe where that surface is. The distinction matters because two very different deformation patterns can produce an identical reading at the top of a structure.

In a rigid body rotation, every point on the structure rotates through the same angle and the displacement increases linearly with height. In distributed bending, the base stays fixed, the rotation increases progressively towards the top, and the displacement profile is curved. A sensor at the top reports one angle in each case. If the two rotations happen to match, the data are indistinguishable from the single reading, yet one structure is rotating as a unit and the other is bending, and the two have different implications for where damage will appear.

A tilt sensor gives you a tangent direction. Turning a set of tangent directions into a displacement profile requires integration, and integration requires two things that a single sensor cannot supply: measurements at several positions along the structure, and a reference point whose displacement is known. Both are engineering decisions, not sensor specifications, and section 6 deals with the second one.

 

4. Which Datasheet Line Actually Limits Your Displacement Reading

A high-accuracy structural monitoring tilt sensor such as the ZCT2xxM-LBS-Ax-H5-460x publishes four separate angular figures, and they behave very differently once multiplied by a span. The datasheet values are a resolution of 0.001 degrees, an accuracy of plus or minus 0.005 degrees typical and plus or minus 0.01 degrees maximum across the measuring range, a zero deviation of plus or minus 0.05 degrees, and a zero temperature drift of plus or minus 0.002 degrees per degree Celsius typical with a maximum of plus or minus 0.13 degrees over the full minus 40 to plus 85 degrees Celsius operating range. As the datasheet notes, unless stated otherwise the tabulated parameters are typical values at room temperature of 25 degrees Celsius.

Converting each of those into millimetres gives the following error budget. Every figure below is a movement in millimetres, and the span is the multiplier that decides which term dominates.

Error term and datasheet value	1 m span	5 m span	10 m span	20 m span
Resolution, 0.001 degrees	0.02 mm	0.09 mm	0.17 mm	0.35 mm
Accuracy, typical, 0.005 degrees	0.09 mm	0.44 mm	0.87 mm	1.75 mm
Accuracy, maximum, 0.01 degrees	0.17 mm	0.87 mm	1.75 mm	3.49 mm
Zero deviation, 0.05 degrees	0.87 mm	4.36 mm	8.73 mm	17.46 mm
Zero temperature drift, worst case, 0.13 degrees	2.27 mm	11.34 mm	22.69 mm	45.38 mm

This is the inversion that most procurement tables miss. The number quoted first in every product comparison is accuracy, and at a ten metre span accuracy contributes under one millimetre. The zero deviation contributes 8.7 millimetres and the worst-case thermal drift contributes 22.7 millimetres, roughly ten and twenty six times the accuracy term. A sensor described as a 0.005 degree instrument is, if read as an absolute displacement reference, a twenty millimetre class instrument over that span.

That is not a defect in the sensor, and it is not a reason to select a different one. It is a statement about which quantity belongs in which question. Resolution and accuracy describe how well the instrument reads the angle in front of it. Zero deviation and thermal drift describe how far the instrument’s zero can sit from the true horizontal, which is a property of the whole signal chain including the mounting. The two groups are used for different purposes, and the next three sections show how the large terms are managed rather than accepted.

 

5. Reconstructing a Profile by Segmental Summation

The standard method for rebuilding a displacement profile from angle measurements is segmental summation, and it is the same arithmetic whether the sensors sit in a borehole casing or on the face of a structure. The element is divided into segments of known length, a sensor measures the rotation of each segment, the rotation of each segment is converted into a displacement increment, and the increments are summed from the reference end.

Two equations describe the whole procedure: the increment across segment i equals the segment length L multiplied by the tangent of that segment’s rotation, and the cumulative displacement at the end of segment i equals the sum of the increments from the reference to that point. Instrument manufacturers publish the same relationship in sine form, which as noted in section 2 is numerically identical at these angles.

A worked example makes the accumulation visible. Take a ten metre element divided into five segments of two metres each. The measured rotations from bottom to top change by 0.01, 0.03, 0.03, 0.02 and 0.02 degrees. Each increment is 2000 millimetres multiplied by the tangent of its angle, giving 0.35, 1.05, 1.05, 0.70 and 0.70 millimetres. The cumulative displacement at each successive joint is then 0.35, 1.40, 2.44, 3.14 and 3.84 millimetres, and the top of the element has moved 3.84 millimetres relative to the fixed base.

Two properties of this method are worth knowing before it is used to set an alarm threshold. The first is that the accuracy of the reconstruction improves as the segments get shorter relative to the curvature of the deformation, because a discrete sum approximates a continuous integral. Published work on inclination-based deflection measurement reports that when the number of segments matches the number of curves in the deformation, the relative error between the calculated and actual value falls below about five percent. The second is that the summation transmits error as well as signal: an uncorrected offset in one segment propagates into every value above it, which is why section 7 is not optional.

Segment length is therefore a design parameter. Continuous in-place systems commonly use half metre segments, which is the minimum sensor spacing published for MEMS in-place inclinometer strings, with one metre and longer segments available where the expected deformation is gentle. Portable probe surveys typically read at half metre depth steps for the same reason.

 

6. Relative or Absolute: The Reference Node Decision

Integration needs a starting value. In practice that means choosing a node whose displacement is assumed to be zero, and everything reported by the profile is measured from it. This choice is an engineering judgement, and it changes the meaning of the result rather than merely its offset.

In a borehole installation the reference is normally the bottom of the casing, on the assumption that it is anchored below the zone of movement. In a bridge pier the base is often taken as fixed if the foundation is considered sound, while in a tall slender structure the top is sometimes taken as the stable point when the crown is known to be held in position. Each of these assumptions is defensible on some sites and wrong on others, and when the assumed reference is itself moving, the reconstructed profile is a relative profile: it shows the shape of the deformation correctly but not its absolute magnitude.

Recovering absolute displacement therefore requires an independent external reference. A satellite positioning station, a levelling benchmark, or a total station survey tied to a stable datum can supply the missing displacement at the reference node, after which the summed profile becomes absolute. The practical consequence for monitoring design is that the assumption behind the reference node should be written down in the monitoring plan and reviewed when the data start to move, because a profile that appears to show no deformation is sometimes a profile in which the reference is moving with everything else.

 

7. Incremental Readings Cancel the Terms You Cannot Control

The two largest terms in the error budget of section 4 are not random noise. Zero deviation and thermal drift are stable offsets: they persist between readings and they do not average out. That property is exactly what makes them manageable, because a stable offset disappears from a difference.

Monitoring is a difference measurement, not an absolute one. The quantity of interest is how much the structure has rotated since a chosen baseline, so the reported angle is the current reading minus the baseline reading at the same location. The zero deviation, being common to both readings, cancels. This is standard practice and it is written into the published interpretation guidance for manual inclinometer surveys, which states plainly that all displacement calculations are relative to an initial reference measurement, and which recommends collecting at least three independent survey readings before selecting a baseline so that drilling disturbance, casing settlement and grout curing can be identified and excluded.

The same logic applies to the thermal term, but with a condition attached. The drift cancels completely only when the two readings being compared are taken at the same temperature. Comparing a January survey with an August survey leaves the residual in place, and over the full operating range of a structural sensor that residual can be, as the budget table shows, larger than everything else combined. Three defences are available and they are normally combined: compare readings taken in the same temperature window, record temperature alongside every reading so that the relationship between angle and temperature can be fitted and removed, or specify an instrument whose thermal behaviour has already been characterised across its temperature range.

This reframes what the selection specification should ask for. If the deliverable is a change in displacement rather than an absolute one, the figure that governs the quality of the answer is repeatability, the spread of repeated readings taken under identical conditions, together with the stability of the zero over time. Absolute accuracy still matters, because it sets how well the profile can be tied to an external datum, but it is no longer the term that decides whether a millimetre of movement is visible.

 

8. What the Profile Shape Tells You About the Mechanism

Once a profile has been reconstructed, its shape carries information that the individual angle readings did not. A plot with position along the structure on one axis and reconstructed displacement on the other produces a curve for each measurement time, and the evolution of that curve is the record of the deformation.

Profile shape	Deformation mechanism	What to check
Straight line	Rigid body rotation of the element	Whether the assumed hinge or rotation centre is correct
Smooth curve	Distributed bending along the length	Curvature distribution and comparison with design assumptions
Sharp inflection point	Localised stress concentration, plastic hinge or sliding surface	Condition survey at that elevation; this is the location to instrument more closely
Step or offset between two portions	Structural or geotechnical discontinuity	Joint condition, or a slip plane across the casing

Overlaying the profiles on environmental records separates the drivers. A profile that grows through a hot afternoon and returns at night is thermal. A profile that only advances during rainfall or during an excavation stage is construction or groundwater related. A profile that advances steadily with no environmental correlation is the one that warrants action. Comparing successive profiles rather than absolute positions also isolates the incremental deformation within a defined interval, which is usually the quantity compared against a rate based alarm threshold.

 

9. What Tilt-Based Displacement Cannot See

A deflection profile built from tilt measurements has definite blind spots, and stating them is part of using the method correctly. Four of them recur.

Uniform settlement is invisible. If an entire structure lowers by the same amount, every segment remains parallel to its neighbours, every measured rotation stays at its baseline value, and the sum of the increments remains zero. The method reports no deformation at all while the structure is settling as a block. Detecting this requires a settlement measurement, not a tilt measurement, and any monitoring plan that relies on tilt alone should say so.

Fast dynamic events are missed. Chain and string systems sample at intervals measured in minutes, which is appropriate for slow deformation and useless for seismic response or the passage of a train. Those phenomena need accelerometers or high rate instruments working alongside the tilt system.

Movement outside the measured plane is only partly captured. A one dimensional chain reconstructs the profile along its own axis. Rotation components normal to that axis require biaxial sensors working together with an analysis that accounts for both directions, and the resultant rotation of a biaxial sensor is obtained by combining the two channels rather than by reading either one alone.

Absolute position is not available without external help, as section 6 set out. The profile is only as absolute as the reference node it was integrated from.

 

10. Field Checklist for Tilt-Based Displacement Monitoring

The following points, in the order they arise on a project, cover the failure modes that account for most disappointing results.

Confirm the range against the mechanism. Structural monitoring rotations are usually well inside one degree, so a plus or minus 15 degree range leaves a large margin, while overturning or tracker applications may need wider coverage. Confirm the range before ordering, because range and accuracy are traded against each other in MEMS designs.

Confirm the mounting orientation against the model variant. ZC Sensor publishes horizontal and vertical variants within the same family, designated AH and AV, and the datasheet states that installing a unit in the wrong orientation affects accuracy. The measurement direction diagram for the specific variant being installed is the reference, not a generic drawing.

Observe the two mounting rules that the datasheet sets out, which it calls the two sides rule and the two lines rule. Two sides means that the mounting surface of the sensor sits completely against the mounting surface of the object being measured, with that surface as horizontal or as flat as possible, so that no angle is generated between the two. Two lines means that the axis of the sensor is parallel to the axis of the surface being measured, so that the two axes cannot be angled relative to one another. The datasheet is explicit that improper installation can result in large angle measurement errors, and this is the most common cause of a profile that drifts without any real movement.

Establish the baseline properly. Take at least three independent readings after installation and before the monitoring period begins, review them for consistency, and discard any that reflect installation disturbance. Record the assumed reference node and its justification in the monitoring plan.

Record temperature at every reading. Either compare readings taken in the same temperature window or fit and remove the thermal relationship. Where continuous strings are used, per segment thermistors make the second option straightforward.

Check the practical electrical figures against the logging plan. For the ZCT2xxM-LBS-Ax-H5-460x these are a supply of 8 to 36 volts DC, a quiescent current of 15 to 20 milliamperes at 24 volts, a refresh time of 40 milliseconds in question and answer mode at 9600 bps, and a power-on startup time of 0.5 seconds. The datasheet note on refresh time is worth applying: consecutive angle queries should be spaced at least 40 milliseconds apart from the end of the previous query, which sets the fastest polling interval the logger can use.

 

Frequently Asked Questions

Q1: What is the formula for converting a tilt reading into a displacement? Displacement equals the span or segment length multiplied by the tangent of the rotation angle. At the small angles used in monitoring, the tangent, the sine and the angle expressed in radians give the same result to within a few parts per million, so the choice of form is not significant. The useful constant is that one degree equals 17.46 millimetres per metre, which means a rotation of 0.005 degrees corresponds to 0.87 millimetres of movement over a 10 metre span.

Q2: How accurate is a tilt sensor when the answer is required in millimetres? It depends entirely on the span, because the span is a multiplier. For the ZCT2xxM-LBS-Ax-H5-460x, the accuracy term of plus or minus 0.005 degrees typical and plus or minus 0.01 degrees maximum becomes 0.87 and 1.75 millimetres over a 10 metre span. The zero deviation of plus or minus 0.05 degrees becomes 8.73 millimetres over the same span, and the worst-case thermal drift of plus or minus 0.13 degrees becomes 22.69 millimetres. If the deliverable is an absolute displacement, the last two terms set the practical floor; if the deliverable is a change since a baseline, they cancel and the governing figure becomes repeatability.

Q3: How many tilt sensors are needed to reconstruct a deflection profile? Enough to approximate the curvature of the expected deformation, which means dividing the element into segments short enough that the rotation within each one is effectively constant. Accuracy improves as segments shorten, and published work reports relative errors below about five percent when the number of segments is matched to the deformation pattern. Continuous in-place MEMS systems commonly use half metre segments, which is the minimum published sensor spacing for that class of instrument, with longer segments where the deformation is gentle.

Q4: Is zero temperature drift more important than accuracy? For a displacement deliverable over a long span, yes, because it is the larger term by an order of magnitude. But the comparison is not really between two specifications; it is between two different questions. Accuracy describes how well the instrument reads the angle in front of it. Zero deviation and thermal drift describe where the instrument’s zero sits, and they are stable rather than random. That stability is why monitoring against a baseline removes them, and why the specification for a monitoring project should weight repeatability and zero stability alongside accuracy.

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