Bridge deflection evaluation through the use of tiltmeters
In this article we will describe the method used to calculate the lowering of a span structure such as a bridge or an overpass starting from the data recorded by the Move tiltmeters positioned on the structure itself.
A little bit of theory
The rotation can be expressed as a function of the lowering by defining its primary derivative, considering the small amount of displacement, as shown in the following equation:
The displacement function can be expressed by utilising a polynomial function for each of the spans of the bridge. The solution is calculated by enforcing generic boundary conditions for the bridge, assuming its behaviour to be equivalent to a beam resting on the masts and supports at each end.
For a generic span j, if n + 1 boundary conditions are known, then the polynomial that interpolates the lowering vj(x) is a function of degree n, with the following mathematical formulation:
Where Cp is the p-th coefficient of the polynomial. For example, an order 3 polynomial would be:
The boundary conditions are obtained from the zero vertical displacement on the supports and from the rotations read by the sensors.
The polynomial function is obtained by finding the coefficients Cp of the polynomial through the simple solution of a system.
A procedure for carrying out the static load test on the structure is shown below.
Step 1 — Sensors positioning
Being L the span lenght, a minimum of 3 sensors must be positioned on 0 (support), 0.15L, 0.3L and other 3 sensors at 0.70L, 0.85L, L (other support). If other sensors are available, they should be positioned equally distributed between 0.3L and 0.7L. For each load test it is necessary to record the exact position of the sensors along the length of the span. Then build a table like the following:
Step 2 — Unloaded bridge
Before starting the loading phase, it is essential to sample with the tiltmeters on the unloaded bridge, for about 5–10 minutes. Wait until the value read by the sensors stabilizes. After being sure of that by observing the values on the web platform, write down the current time on the table on “Time of unloaded bridge”.
Step 3 — Loaded bridge
Once the load has been positioned, record the time when the bridge was loaded (“Time of loaded bridge”). Wait for the bridge to deform and lower. Once the inclinometers values on the platform stop changing, it means that the deflection value is stable, therefore record the “Time of end measuring” on the table.
Perform this procedure for all spans.
Step 4 — Deflection extrapolation
For the calculation of the deflection we have to apply some basic concepts of linear algebra. Each tiltmeter provides the rotation of a point of the deflection. For example, let’s assume that we have 3 tiltmeters and therefore 3 rotation points of the deflection.
The 3 rotation points are part of the boundary conditions (as explained above). We also have two other boundary conditions, the lowerings near the supports equal to 0.
So, in the end, we have 5 boundary conditions, given by the 3 inclinometers plus the 2 supports, and therefore our polynomial will be of order 4, since:
So the interpolating polynomial will be the following:
We have to set up some linear equations where the known terms are given by the boundary conditions.
The first two equations come from the boundary conditions of the supports:
Now we need the equations with the information provided by the tiltmeters.
We assume now, that during the load tests the positions of the tiltmeters were the following:
The measurement of rotation of the inclinometers is exactly the derivative of the lowering, therefore we have to set up 3 equations based on the derivative of the interpolating polynomial, which is the following:
The third equation is set with sensor 1 in position 0:
The fourth equation is set with sensor 2 in the 0.5L position:
The fifth equation is set with sensor 3 in position L:
At this point we just have to solve the system to find the coefficients of the interpolating polynomial.
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Move Solutions design and produces complete structural health monitoring systems for civil works (SHM), developing wireless sensors networks with LoRaWAN communication.
Find out more on www.movesolutions.it or contact us at info@movesolutions.it.
