Evaluation of Photogrammetric Methods for Displacement Measurement During Structural Load Testing

Abstract

The safety and longevity of engineering structures depend on precise and timely monitoring, especially during load testing inspections. Conventional displacement measurement methods—such as LVDT sensors, GNSS, RTS, and levels—each present benefits and limitations in terms of accuracy, applicability, and practicality. Photogrammetry has emerged as a promising alternative, offering non-contact measurement, cost-effectiveness, and adaptability in challenging environments. This study investigates the potential of photogrammetric methods for determining structural displacements during load testing in real-world conditions where such approaches remain underutilized. Two photogrammetric techniques were tested: (1) a single-image homography-based approach, and (2) a multi-image bundle block adjustment (BBA) approach using both UAV and tripod-mounted imaging platforms. Displacement results from both methods were compared against reference measurements obtained by traditional LVDT sensors and robotic total station. The study evaluates the influence of different camera systems, image acquisition techniques, and processing methods on the overall measurement accuracy. The findings suggest that the photogrammetric method, especially when optimized, can provide reliable displacement data with sub-millimeter accuracy, highlighting their potential as a viable alternative or complement to established geodetic and sensor-based approaches in structural testing.

1. Introduction

Engineering structures such as buildings, bridges, and dams require continuous monitoring during construction and throughout their lifespan to ensure safety and functionality. Over time, materials deteriorate, reducing load-bearing capacity and performance. Structural inspections can be routine, periodic, post-event damage assessments, or focused on specific components with known vulnerabilities. Structural health monitoring (SHM) uses real-time sensor data to detect early signs of distress, while load testing verifies strength and stability by applying controlled loads. Both static and dynamic displacement measurements are essential for understanding structural behavior, with static monitoring assessing long-term deformations and dynamic monitoring capturing vibrations or short-term effects from loads such as traffic, wind, or seismic events.

Determining displacement during a load testing inspection can be carried out using various methods and instruments. When selecting measuring instruments during the planning phase of the measurement, attention should be paid to the accuracy of measurements, ease of conducting the load testing, and cost-effectiveness. The selection of instrumentation is left to the expert and will depend, among other factors, on the expected displacement values of the structure, site conditions prevailing at the location of the load testing, and accessibility to the measurement points on the structure or the possibility of placing sensors. Contact sensors such as Linear Variable Differential Transformers (LVDTs) are traditionally used to measure one-dimensional displacements in a single plane during structural testing [1,2,3,4,5], with typical accuracy to ±0.1% of full-scale output. Although these sensors can provide high accuracy in measuring relative displacements between a measurement point on the structure and a reference point independent of the structure being tested, a limitation is that it is often difficult to establish a fixed reference point, especially when testing difficult-to-access structures such as bridges spanning rivers or roads, or structures that are high above the ground, etc.

In such cases, displacements can be determined by using classical geodetic non-contact instruments such as GNSS instruments, robotic total stations (RTS), and levels. To determine vertical displacements during load testing, levels are extensively used, providing high-precision measurements [6,7,8]. From the measured height difference, the displacements can be calculated with an accuracy of 0.1 mm. Although the use of levels represents a reliable and highly precise method for determining vertical displacements, a disadvantage of the method is that it is time-consuming and not suitable for continuous monitoring projects, and it can only determine static vertical displacements of structures. Due to their advantages, GNSS instruments are widely used to measure both static and dynamic displacements of large structures such as high-rise buildings, towers, dams, and bridges [8,9,10,11,12,13]. Static displacements can be measured using the GNSS relative static positioning method with millimeter-level accuracy [14], while dynamic displacements can be measured using relative positioning methods such as Real Time Kinematic (RTK) and Post-Processing Kinematic (PPK), as well as the absolute positioning method Precise Point Positioning (PPP), with centimeter to sub-centimeter-level accuracy [15,16,17]. By applying the PPP-GNSS method, displacements can be determined using only a single instrument, making this method more cost-effective compared to relative GNSS positioning methods, as there is no need to install a reference instrument outside the structure being observed. The achievable accuracy in determining displacements using GNSS technology is significantly influenced by the number of visible satellites and their geometry. In order to improve GNSS positioning accuracy in areas with obstructions to signal propagation and to enhance the vertical component accuracy, which is typically two to three times worse than the horizontal one, the use of pseudolites improves the constellation of available GNSS satellites and enhances the accuracy of displacement measurements, achieving similar accuracy levels (sub-cm) in both horizontal and vertical planes [14,18,19,20]. In structural dynamic monitoring, integrating GNSS with accelerometers provides a more comprehensive view of structural behavior, as GNSS is less effective for small-amplitude displacements, while accelerometers are limited in capturing low-frequency movements [10,14,15].

RTS instruments are also used in combination with GNSS technology in projects, but they can also serve as a standalone method for determining static and dynamic displacements of structures [1,3,21,22,23,24,25]. With modern RTS, sub-millimeter accuracy in determining vertical displacements can be achieved [6,26]. One of the advantages of RTS as a more cost-effective solution in continuous monitoring projects, where displacements need to be measured at multiple points, is that it requires only a single instrument to measure towards reflectors placed at the points, unlike GNSS, which needs a separate instrument at each measurement point. Robotic total stations, like GNSS instruments, can also be used for the dynamic testing of structures [27,28,29,30], with the ability to determine dynamic displacements at only one measurement point and a limited measurement frequency of up to 20 Hz [7,28].

Additional capabilities of RTS in monitoring projects, as well as in determining static and dynamic displacements of structures [31,32], are enabled by integrating imaging sensors into the instrument’s telescope. Such instruments are called Image-Assisted Total Stations (IATS). The paper of [33] demonstrates the application of an IATS prototype for monitoring dynamic displacements in the laboratory and during load testing of a railway bridge. The imaging sensor allowed the identification of a third bridge frequency of 15.31 Hz, which would not have been possible with the RTS due to insufficient measuring frequency. To achieve higher measurement accuracy for static vertical displacements, the authors of the paper in [34] demonstrated the capabilities of IATS when measuring on a digital leveling staff placed at the measurement location instead of a reflector. Based on tests conducted in the laboratory with measurement distances ranging from 5 to 15 m, the authors achieved identical precision with the IATS as with a precision level. The capabilities of IATS in measuring simulated dynamic vertical displacements on a digital leveling staff were presented in the paper [35].

In addition to these instruments, terrestrial laser scanners (TLS) are also intensively used in structural testing projects, providing highly precise measurements and 3D modeling of load-bearing structures under different loads [36,37,38]. This measurement method provides a detailed insight into the geometry of a structure, including any deformations or damage that might be difficult to detect with conventional methods. The advantage of TLS over GNSS and RTS is its ability to collect very high data density on the object of interest in a relatively short time without the need for physical contact with the structure. Additionally, TLS can provide accuracy in the millimeter range, which is more precise compared to GNSS accuracy, and it reduces the time required for field measurements, thus improving the efficiency of the measurement process. In the paper [39], the authors analyzed the achievable precision of TLS based on results from two load tests of the bridges and compared the results with a precise level result. The authors note that TLS cannot completely replace high-precision traditional geodetic instruments in projects where high accuracy is required, but they highlight the advantages of using the instrument as an additional technology since it provides valuable 3D information covering the entire surface. Similarly, the authors in the paper [40] achieved vertical displacement measurement accuracy during load testing of a railway bridge with an accuracy of ±1 mm and point out the benefits of using TLS as additional measurements alongside precise leveling in these projects.

Authors increasingly highlight the possibilities and advantages of using cameras and photogrammetric methods in load testing and structural monitoring projects [41,42,43,44,45,46,47,48,49,50]. Vision-based approaches are attractive because they rely on low-cost and easily deployable camera systems, while different image-processing techniques—such as Digital Image Correlation (DIC), Structure-from-Motion (SfM), and Multi-View Stereo (MVS)—enable non-contact displacement measurements in either a 2D or a full 3D space. These methods can be deployed without pre-installed targets, which is particularly advantageous in hard-to-access or sensitive locations, and allows for the monitoring of static or dynamic displacements, as well as surface strains and rotations [47,48]. Several studies have demonstrated the potential of DIC-based monitoring, achieving sub-millimeter accuracy in mid-span deflection measurements during static load testing of railway bridges [49] or integrating UAVs with DIC for efficient bridge testing [50]. Similarly, UAV-based SfM/MVS workflows have proven effective for dense area-wide deformation monitoring: the authors in [51] showed that high-resolution UAV photogrammetry, with a ground sampling distance of 1.3 mm, achieved less than 1 mm deviation compared to reference displacement transducers, enabling a full 3D description of bridge bending behavior under static loads. The authors in [52] further demonstrated that UAV flight altitude, overlap configuration, and ground control point distribution significantly influence the accuracy of 3D reconstructions, with achievable millimeter-level precision in controlled bridge deformation tests. Such computer-vision-based techniques, therefore, complement traditional geodetic instruments by offering flexible, large-scale deformation monitoring at lower cost and setup effort.

Table 1. Characteristics of the instruments and methods used for determining static displacements of structures.

Method/Instrument	Advantages	Disadvantages	Accuracy	Measured Parameters	Application Examples
LVDT	High precision, real-time displacement tracking, automated measurements, unaffected by weather conditions	Requires contact with the structure. Setting a reference point can be challenging. Limited application for large displacements.	Up to 0.01 mm	Displacement (1D)	Static load testing of the bridge [3], Load testing of the roof structures [2]
GNSS	High precision in static measurements, real-time displacement tracking, automated measurements, capable of long-distance measurements	Requires contact with the structure. Depends on satellite visibility. Multipath effects reduce accuracy. Less accurate in vertical than horizontal plane.	Sub-cm (static), cm (dynamic)	Displacement (vertical, horizontal)	Static load testing of the bridge [17,53,54,55]
RTS	Sub-millimeter accuracy, automated measurements, real-time displacement tracking, cost-effective for multiple static measurement points	Requires contact with the structure. Requires line-of-sight. Sensitive to weather (fog, rain, wind). Higher cost compared to simpler instruments	Sub-millimeter accuracy	Displacement (vertical, horizontal)	Static load testing of the bridge [3,54,56,57], Static load testing of the bridge pier [58], Static load testing of foundation piles [59], Static load testing of the beam [6]
Level	High precision, lower cost compared to advanced instruments	Only measure vertical displacements. Not suitable for dynamic displacement. Sensitive to weather (fog, rain, wind). Impossibility of automated displacement measurement process	0.1 mm	Displacement (vertical)	Static load testing of the bridge [3,38,53,60]
TLS	High data density, fast and accurate 3D surveying over large areas, non-contact method	Less accurate than traditional methods. Requires line-of-sight. Expensive and require expert data processing. Sensitive to weather (fog, rain, wind).	1–2 mm	3D displacement, object geometry	Static load testing of the bridge [36,38,39,56,57,60,61,62,63], Static load testing of foundation piles [59], Static load testing of the beam [6,64]
Photogrammetry	Low cost, fast and simple data collection, non-contact method, long-range capability	Lower accuracy than high-precision methods. Requires line-of-sight. Affected by lighting and weather (fog, night). Requires post-processing and data analysis.	Sub-millimeter accuracy	Displacement (in sensor plane or 3D)	Static load testing of a concrete block [65,66], Static load testing of the bridge [49,67,68], Static load testing of the beam [69]

summarizes the main instruments and methods for measuring static displacements during load testing and structural monitoring. It compares their accuracy, advantages, and limitations, providing guidance for selecting suitable techniques under different site conditions.

The aim of this study is to investigate the application of the photogrammetric method for measuring displacements in a load testing of structures. This method offers significant advantages in structural testing, especially when photogrammetry provides a more cost-effective solution or when other methods are impractical. Photogrammetry, particularly when combined with UAV technology, has gained increasing popularity in various fields due to its ability to rapidly capture large-scale areas with minimal setup, while maintaining a high level of precision. Unlike traditional methods that require direct contact with the structure, photogrammetric approaches enable non-invasive monitoring, which can be especially useful in sensitive or hard-to-reach areas. This study will assess the potential of UAV-based and camera-based photogrammetry to deliver reliable displacement measurements, comparing the accuracy, ease of implementation, and practical limitations of photogrammetry against the more conventional LVDT sensor-based approach.

Due to all the above, in this study, we tested two different approaches to the photogrammetric method. In the first approach, we used only a single image for each loading phase of the beam, using the mathematical model of homography to determine the displacements between characteristic points marked by photogrammetric signals. In the second method, we used a photogrammetric series of images (an image strip) taken once from fixed locations on the ground and a second time using a drone as a recording platform. The final 3D coordinates of the characteristic points for each loading phase were determined using the Bundle Block Adjustment method (BBA). The results will be compared with traditional reference values obtained using LVDT displacement sensors. This paper aims to evaluate the accuracy of different displacement measurement methods, focusing on the achievable precision of different approaches of the photogrammetric method (pure homography and BBA), different imaging platforms (tripod-mounted camera, UAV-borne camera) and different cameras used (full-frame NIKON D800E, full-frame NIKON D750, DJI FC6310S), all belonging to the consumer class.

2. Materials and Methods

To verify the applicability of photogrammetric method for displacement measurements in load testing of structures, by two different approaches using cameras on UAVs and tripod-mounted cameras, a test was conducted on a prestressed reinforced concrete beam with a length of 36 m. The beam has a two-way slope of 2%, with a total height of 144 cm at the ends and 180 cm at the center. The width of the upper flange is 74 cm, and its height is 30 cm, while the width of the web is 25 cm (Figure 1). In order to ensure the repeatability of the test, the investigation was conducted in accordance with the national standard HRN U.M1.047 [70] for testing high-rise building structures using load testing and load-to-failure procedures. The standard requires the observation of displacements at relevant points during the various phases of loading and unloading the structure.

The loading was applied in stages, with an increment of 25% at each step. The static load test was conducted in four loading phases and four unloading phases to evaluate the beam’s behavior under different stress conditions. For the loading, a total of 24 pallets, each weighing 2.48 tons, were used, resulting in a cumulative load of 59.52 tons (Figure 2). This total load was carefully distributed to simulate realistic conditions and was chosen to ensure that the test would achieve the maximum internal forces expected on the beam according to its design specifications. The described procedure and loading ensure that the displacement measurements on the structure correspond to realistic values that could occur under practical conditions.

The load testing was carried out in the following phases:

• Phase 0: beam unloaded,
• Phase 1: beam loaded with 25% of the pallet (Figure 3a),
• Phase 2: beam loaded with 50% of the pallet (Figure 3b),
• Phase 3: beam loaded with 75% of the pallet (Figure 3c),
• Phase 4: maximum load: beam loaded with 100% of the pallet (Figure 3d),

After the beam was gradually loaded to the maximum load, the load testing of the beam continued with the phases of unloading:

• Phase 5: beam loaded with 75% of the pallet,
• Phase 6: beam loaded with 50% of the pallet,
• Phase 7: beam loaded with 25% of the pallet,
• Phase 8: beam unloaded.

During the load testing, vertical displacements were continuously measured at each phase using an LVDT sensor. In addition, periodic measurements were taken after each loading and unloading phase using a robotic total station, camera with two different displacement measurement approaches, and UAV photogrammetry method. The measurement methods, locations of measurement points, and the instruments used are detailed in the following section of this chapter.

2.1. LVDT Sensors Measurements

During the load testing, displacements were measured with 7 LVDT sensors. Vertical displacements were measured at the center of the span (measurement point 5), as well as at the eighths of the span on the left side of the beam (measurement points 2, 3, and 4), and vertical displacements at the left bearing (measurement point 1) and right bearing (measurement point 9). Additionally, the horizontal displacement of the beam was measured at the center of the span (measurement point 10) (Figure 4).

During the load testing, vertical displacements were continuously measured at each phase using an LVDT sensor. The LVDT sensors used in the measurements were manufactured by HBM, model type WA, with a measurement accuracy of 0.01 mm. These sensors are known for their high precision, making them suitable for accurate displacement monitoring. To ensure the reliability and accuracy of the data, all sensors were carefully calibrated using a standard calibration procedure before the measurements were taken. Given that the LVDT sensors offer significantly higher measuring accuracy compared to other observed methods, the displacements measured by the LVDT were considered the reference values for the study. These reference values were then used to assess and compare the accuracy of other displacement measurement techniques employed in this research.

2.2. Robotic Total Station Measurements

The displacements of the structure during the loading of the beam were determined using a robotic total station Leica TPS1201, with a direction measurement accuracy of 1″ and a distance measurement accuracy of 2 mm + 2 ppm. Displacements were measured in seven monitoring points (points 2–8) located on the beam as shown in Figure 5a. The monitoring points were stabilized in such a way that the prisms were fixed to the lower part of the beam (Figure 5b).

The measurements were carried out using an automated measurement procedure. This has been enabled since the RTSs are equipped with servomotors and automatic target recognition (ATR) functions, which enable precise targeting and sighting on the prism with an automated measurement process. Before each load phase, the RTS station coordinates were obtained based on measurements towards two reference points using the free station method [25,71]. Using this method, the coordinates of the unknown point (RTS station) and their standard deviations can be calculated by least squares adjustment based on the measured slope distance, horizontal direction, and zenith angle toward the two reference points. As shown in Figure 6, reference points with known coordinates (R1 and R2) are set near the beam in the stable area, while the RTS is in front of the carrier at a distance of about 22 m.

After the RTS station coordinates are calculated, the coordinates of the seven measurement points (points 2–8) are determined by the polar method [25].

X_i = X_RTS + d_i × sinz_i × cos(φ₀ + φ_i)
Y_i = Y_RTS + d_i × sinz_i × sin(φ₀ + φ_i)
Z_i = Z_RTS + d_i × cosz_i

(1)

where X_i, Y_i and Z_i are coordinates of the monitoring point i, φ₀ is azimuth of total station’s zero direction, and d_i, φ_i and z_i are observations of slope distance, horizontal direction, and zenith angle from the RTS toward each monitoring point.

The 3D coordinates of the monitoring points 2–8 were determined after every loading phase in the local coordinate system with the X-axis parallel to the beam, Y-axis perpendicular to the X-axis in horizontal plane and Z-axis perpendicular to the horizontal plane (the plane defined by the X- and Y-axes) In this way, changes in X-value of monitoring points are displacements in longitudinal direction of the beam, changes in Y-value displacements in lateral direction of the beam, and changes in Z-direction represent vertical displacement.

Due to the short measuring distance (max. measuring distance was 31 m) and insignificant temperature changes during the load testing, corrections of the measurements for the earth’s curvature and the influence of refraction were negligible and were not applied.

2.3. Photogrammetric Measurements

Displacements of the beam during the load testing were determined using the photogrammetric method, applying two different approaches. In both approaches, displacements were determined at 18 points marked with black circular targets of 8 cm radius (Figure 7).

Displacements of the beam using the photogrammetric method were determined at 9 points (points 1–9 in Figure 7). Points stabilized on the beam supports and on the ground beneath the beam (points 10–18) represented reference points relative to which the displacements of the points on the structure were determined. Reference points 14 and 15 were damaged during the loading phases as a result of the beam settling and were not used as reference points in data processing.

2.3.1. Homography of a Single Terrestrial Image (HSTI)

As part of the first approach, the measurement of the beam’s displacement was carried out from a single camera viewpoint, meaning that the displacements were determined based on one photograph that included both ground control points (GCP) and check points on the beam itself. We chose a digital camera, NIKON D800E, with a SIGMA lens (f = 20 mm, f/D = 1.8) to photograph the beam.

To achieve optimal accuracy in determining the beam displacement, it was necessary to establish the measurement potential of the digital camera intended for conducting the displacement measurements before the actual measurement. This measurement potential is expressed as the cutoff frequency of the sample that can still be recorded by the camera, and it is displayed on the modulation transfer function (MTF) diagram. Along with the MTF, the number of line pairs per millimeter that the camera can still resolve is traditionally used as a measure to determine the measurement potential. Beam displacements are monitored on photogrammetric signals of circular shape to enable high measurement accuracy using fully automated image correlation methods (least squares matching—LSM). This feature enables the measurement accuracy of the image coordinates to range from 1/4 to 1/10 of the pixel size. The measurement features of the NIKON D800E camera and the SIGMA RF20/1.8 lens were studied by analyzing the modulation transfer function (MTF) on a test chart (Figure 8).

The MTF diagram in the central position shows a very high resolving power of the lens used. If we can accurately detect and measure the signal at 20% of the original contrast, then in the center of the image (the blue dot in Figure 8 and the blue diagram) we can expect a maximum resolving power of 194.06 line pairs per millimeter (lp/mm), which means that there is a minimum distance of 5.1 μm between adjacent line pairs. In the corners of the image (orange dot and diagram), the resolving power is about half as high at 98.8 lp/mm or 10.1 µm for a line pair. Since the beam extends mainly in the horizontal direction and occupies a relatively narrow strip on the image in this direction, it is useful to look at the resolution at the side of the image (green dot and diagram), where it is 163.12 lp/mm or 6.1 um for a line pair. We used this value to calculate the a priori accuracy of the photogrammetric measurement. The condition for photographing from a single viewpoint is that the entire beam is captured in one shot. The beam is about 36 m long, and the horizontal dimension of the sensor is 36 mm. Therefore, it is necessary to photograph at a scale of 1:1000, which gives us theoretical pixel dimensions of 4.89 mm on the beam. However, based on the study of the actual spatial resolution using the MTF, we find that to image a pair of lines we need to capture 6.1 µm in the image, which gives us a size of 6.1 mm on the beam and an expected a priori measurement accuracy (for circular signals by automatic correlation) of 0.25 × 6.1 = 1.5 mm. The camera is mounted on a stable tripod and triggered with a remote shutter release to prevent the camera from shaking while taking a picture (Figure 9).

In each loading phase of the beam, 10 photos were taken with image size of 7360 × 4912 pixels. This allowed an assessment of the accuracy in determining the coordinates of the signaled points using this approach, as the points remained stationary during each measurement phase—the load was static. A total of 90 photos were taken, i.e., 10 photos per load phase. The time required to take 10 photos was less than 2 s, which can be emphasized as an advantage of this measurement method. Since the entire test process, which included nine loading phases, lasted several hours, a homography transformation procedure was performed to eliminate the influence of possible instability of the camera over time. The pixel values of the ground control points (points 10–13) are used to create the homography transformation between the image planes of the first phase and the other loading phases. Then the pixel values of the points on the beam (points 1–9) and check points stabilized on the ground beneath the beam (points 12, 13 and 16) were corrected by applying the homography transformation. Since all the signaled points were in the same plane, it was possible to apply this correction procedure, which ensures the consistency of the image analysis for all loading phases.

2.3.2. Bundle Block Adjustment of Terrestrial Images (BBA-TI)

In contrast to the first approach, in which the displacements of the points on the beam were determined from a fixed camera position based on a single photograph that included both the reference points and the points on the beam, the second approach used photogrammetric imaging performed in a single sequence. It involved a total of 26 camera positions placed 1.4 m apart along a line parallel to the beam, at a distance of 10 m. In total, 18 points were marked with photo signals on the beam, of which the deformation of the beam was monitored at 9 points (points 1–9), while the remaining points were used for orientation and pre-calibration of the array. The coordinates of all points were determined using the Agisoft Metashape Pro v.1.7.4 software during the image alignment phase. It includes structure from motion (SfM), aerial triangulation (AT) and bundle block adjustment (BBA). At this stage, Metashape searches for feature points on the images and matches them across images into tie points. The program also finds the position of the camera for each image and refines camera calibration parameters (estimates internal (IO) and external (EO) camera orientation parameters).

Circular signals were observed on all images using an automated image correlation procedure, as implemented in Agisfot MetaShape software, using the Least Square Matching (LSM) method. The images were taken with a NIKON D750 camera (Nikon Corporation, Tokyo, Japan) (24 Mpix full frame CMOS sensor) with an AF NIKKOR 50/1.8 lens. The measurement potential of the camera and lens was determined by analyzing the MTF diagram (as for the static image) and it was found that neither the lens nor the image processor significantly degraded the sharpness of the captured photo, so it can be assumed that the digitization interval corresponds to the size of the single photodetector (pixel) in the image plane of the camera.

The configuration for photogrammetric imaging is determined by analyzing the a priori accuracy of a normal case of stereophotogrammetry.

• Image dimensions: 35.9 × 24 mm, 6016 × 4016 pixels
• Dimension of a single pixel in the image: d = 5.97 μm~d = 6 μm

For the normal case of stereophotogrammetry:

$$\begin{array}{c}{\mathsf{\sigma }}_{\mathrm{x}}=\sqrt{{\left({\displaystyle \frac{\mathsf{\xi }}{\mathrm{c}}}{\mathrm{m}}_{\mathrm{b}}{\displaystyle \frac{\mathrm{Y}}{\mathrm{B}}}{\mathsf{\sigma }}_{{\mathrm{p}}_{\mathsf{\xi }}}\right)}^2+{\left({\mathrm{m}}_{\mathrm{b}}\cdot {\mathsf{\sigma }}_{\mathsf{\xi }}\right)}^2}\\ {\mathsf{\sigma }}_{\mathrm{y}}={\mathrm{m}}_{\mathrm{b}}{\displaystyle \frac{\mathrm{Y}}{\mathrm{B}}}{\mathsf{\sigma }}_{{\mathrm{p}}_{\mathsf{\xi }}}\\ {\mathsf{\sigma }}_{\mathrm{z}}=\sqrt{{\left({\displaystyle \frac{\mathsf{\eta }}{\mathrm{c}}}{\mathrm{m}}_{\mathrm{b}}{\displaystyle \frac{\mathrm{Y}}{\mathrm{B}}}{\mathsf{\sigma }}_{{\mathrm{p}}_{\mathsf{\xi }}}\right)}^2+{\left({\mathrm{m}}_{\mathrm{b}}\cdot {\mathsf{\sigma }}_{\mathsf{\eta }}\right)}^2}\end{array}$$

(2)

where X, Y and Z are three axes of object coordinate system, where the origin is in left projection center, X-axis is oriented towards the right projection center, Y-axis correspond to optical axis of the left camera and it is oriented towards the object, Z-axis is oriented up and is orthogonal to X–Y plane.

σ_x, σ_y and σ_z—variances of object coordinates x, y and z, respectively, determined by stereophotogrammetry

ξ, η—measured image coordinates

σ_ξ, σ_η and σ_pξ—variances of measured image coordinates ξ, η and stereoparallax p_ξ, respectively

B—the photogrammetric base, distance between left and right projection center

m_b—denominator of image scale, at a camera distance to the beam of 10 m, then the denominator of the image scale is m_b = 10/0.0500 = 200.

With the image correlation method (LSM) on circular signals, it is possible to achieve a measurement accuracy of 1/4 to 1/10 pix. We can, therefore, assume that the variance is σ_ξ = σ_η = 6/4 = 1.5 μm, which gives us the maximum possible measurement accuracy in the center of the image of 0.3 mm on the beam or at the left and right ends of the image of 0.5 mm. The base ratio can be determined from the formula for σ_y. With an overlap of 80% between the images in the series, each point should be mapped on at least 4 images. The length of the beam shown in one image is 7 m. With an overlap of 80%, we, therefore, need a base size of 1.4 m (20% of 7 m). The base ratio Y/B is then 10/5.6 = 1.79, which gives us a depth determination accuracy of σ_y = 0.54 mm.

Therefore, a distance from the beam of 10 m and a distance between the recording points of 1.4 m was chosen for the recording (Figure 10).

The deformations of long image blocks were considered during the camera calibration phase, where all check points on the beam were used to calibrate the camera, determine the radial and tangential distortion and the discrepancies on these points were analyzed and accepted as suitable for the purpose of photogrammetric measurement. It was performed only before the beginning of beam load testing, when the beam was unloaded. After loading the beam, it deforms itself, and all points change position except those at the very beginning and the very end of beam, which remain at the same position. That is why only these points were used for photogrammetric orientation purposes. The overlap between images was 80% (BBA-TI) and even bigger (BBA-UAV), exactly to lower the impact of bending of the image strip. Other methods, i.e., integrated sensor orientation, or implementing the deep learning and computer vision methods, were not implemented because the training of the deep learning model needs a lot of images and high computing capabilities, which were not acceptable in this project.

2.3.3. Bundle Block Adjustment of UAV Images (BBA-UAV)

The UAV photogrammetry method was used as an additional method for determining the beam displacement.

One of the main advantages of UAV photogrammetric measurements over classical terrestrial methods is their applicability in areas where the terrain is too inaccessible for traditional approaches. UAVs can access such areas from the air and capture images from various angles without the need to set up ground control stations, which is particularly beneficial in hard-to-reach or high-altitude locations. As a result, this method increases efficiency on certain projects and reduces the need for additional equipment and human intervention.

The UAV measurement was carried out using the same approach as the camera measurements by linear imaging (BBA-TI method), which was explained in Section 2.3.2. The DJI Phantom 4 Pro was used for the measurement, a drone with the DJI FC6310S camera with a 1″ CMOS 20 MPix sensor. The dimensions of the sensor are 13.2 × 8.8 mm, and the dimensions of the image are 5472 × 3648 pixels. The nominal size of a single photo detector (pixel in the image) is 2.41 × 2.41 μm. The lens has a focal length of 8.8 mm and a maximum aperture of f/2.8 [72].

As with the NIKON D750 camera, the acquisition configuration is determined by analyzing the a priori accuracy of the normal case of stereophotogrammetry [73]. At a camera-to-beam distance of 10 m, the denominator of the acquisition scale is m_b = 10/0.0088 = 1136, resulting in a pixel size on the beam of 2.7 mm and a maximum measurement accuracy for circular signals in the center of the image of 0.7 mm. To achieve 80% overlap between adjacent images in the image strip, the distance between neighboring locations is 13.2 × 1136 × 0.2 = 2999 mm, approximately 3 m.

During the shooting, the camera was triggered manually, so the obtained sizes of the photogrammetric base are only approximately the same as those calculated and planned.

The good lighting conditions allowed the use of short exposure times (1/400 s) and small apertures (f/10). However, the considerable influence of chromatic aberration deteriorates the image quality (Figure 11) and impairs the accuracy of measurement.

The image coordinates of the circular signals were measured using image correlation according to the least squares method (LSM). The object coordinates on the beam were determined using the Bundle Block Adjustment (BBA) method for the entire image strip, whereby only the Z-component of these coordinates is relevant for monitoring beam deformations due to loads.

Compared to features in variable textures, the image coordinate measurements for circular targets with high contrast are unquestionably higher. However, because the circular targets are situated nearly in a line, their spatial distribution is poor for relative image orientation. For this reason, the features found on the beam and even in its background helped to improve the image alignment quality and the relative orientation’s reliability.

3. Results

The aim of this paper is to examine the accuracy of different displacement measurement methods used in static load testing of structures, with emphasis on the achievable accuracy of the photogrammetric method. Each method has its own advantages and disadvantages, and by knowing the achievable precision and reliability of the measurement results for each method, it is possible to more appropriately select the optimal measurement method for a specific load testing project. For this purpose, the results of the displacement measurements obtained using different methods during the static load testing of the concrete beam are presented and analyzed in this section. The methods applied include continuous high-precision measurements with LVDT sensors, periodic measurements using a robotic total station (RTS), and three photogrammetric approaches: (1) homography of a single terrestrial image (HSTI), (2) bundle block adjustment of terrestrial images (BBA-TI), and (3) bundle block adjustment of UAV-based images (BBA-UAV). Each method offers different levels of precision, operational constraints, and practical applicability in field conditions. In

Table 2. Overview of measurement methods and their characteristics.

Method	Instrument	Image Type/Acquisition	Measurement Distance (m)	Image Resolution (px)	Pixel Size on Object (mm/px)	Notes
LVDT	HBM WA (Hottinger Baldwin Messtechnik GmbH, Darmstadt, Germany)	–	–	–	–	Reference method
RTS	Leica TPS1201 (Leica Geosystems AG, Heerbrugg, Switzerland)	–	~22–31	–	–	Polar method, automated measurements, 3D coordinates
HSTI	Nikon D800E + SIGMA 20 mm (Nikon Corporation, Tokyo, Japan)	Single image, tripod	~22–31	7360 × 4912	~4.9 mm	Homography transformation applied
BBA-TI	Nikon D750 + 50 mm (Nikon Corporation, Tokyo, Japan)	Image strip (26 images), tripod	10	6016 × 4016	~1.2 mm	LSM + BBA with self-calibration
BBA-UAV	DJI Phantom 4 Pro (DJI, Shenzhen, China)	Image strip (manual trigger), UAV	~10	5472 × 3648	~2.7 mm	LSM + BBA with self-calibration

, an overview of the measurement characteristics for each method is summarized.

Table 3. Vertical displacements (mm) measured by different methods.

Point	Method	Phase 1	Phase 2	Phase 3	Phase 4	Phase 5	Phase 6	Phase 7	Phase 8
1	LVDT	0.26	0.68	1.19	1.70	1.69	1.43	1.16	0.77
	RTS	-	-	-	-	-	-	-	-
	HSTI	0.80	1.20	2.60	3.90	2.42	1.32	1.23	0.32
	BBA-TI	-	1.10	-	3.08	-	1.92	-	0.62
	BBA-UAV	-	−0.72	-	3.06	-	2.32	-	2.83
2	LVDT	5.38	11.70	19.16	27.82	22.91	16.66	10.36	4.01
	RTS	5.26	11.36	18.45	26.86	22.07	15.83	9.93	3.85
	HSTI	5.95	11.33	18.78	26.38	23.09	15.60	9.89	5.07
	BBA-TI	-	11.97	-	26.47	-	16.04	-	3.76
	BBA-UAV	-	9.75	-	27.29	-	16.37	-	5.03
3	LVDT	9.33	20.95	33.55	48.55	39.81	28.86	17.91	6.96
	RTS	9.55	20.32	32.95	47.79	38.84	27.92	17.19	6.33
	HSTI	9.97	20.81	33.59	47.74	37.88	27.48	17.97	6.35
	BBA-TI	-	21.23	-	47.46	-	28.25	-	6.21
	BBA-UAV	-	19.94	-	49.38	-	28.92	-	8.24
4	LVDT	11.87	25.84	42.15	61.47	50.26	35.99	22.04	8.10
	RTS	12.27	25.57	41.75	60.81	49.57	35.38	21.5	8.00
	HSTI	12.31	26.05	42.53	61.26	50.20	36.32	22.71	8.87
	BBA-TI	-	26.65	-	60.29	-	35.40	-	7.28
	BBA-UAV	-	24.42	-	62.01	-	35.88	-	9.51
5	LVDT	12.57	27.13	44.35	64.96	53.11	38.87	24.52	9.98
	RTS	13.12	27.59	44.75	65.10	53.02	37.92	23.17	8.66
	HSTI	10.91	28.55	46.26	65.17	53.50	37.78	23.46	8.94
	BBA-TI	-	28.66	-	64.67	-	38.26	-	8.13
	BBA-UAV	-	24.34	-	65.03	-	37.84	-	9.15
9	LVDT	0.01	0.10	0.17	0.28	0.40	0.41	0.40	0.36
	RTS	-	-	-	-	-	-	-	-
	HSTI	0.50	1.60	2.60	3.30	2.70	2.20	3.39	1.50
	BBA-TI	-	0.70	-	1.70	-	1.00	-	0.31
	BBA-UAV	-	−2.80	-	2.00	-	3.11	-	2.21

provides the displacements measured by the above-mentioned methods at six common points along the beam. In this study, downward vertical displacements (i.e., settlement of the structure) are considered positive, while upward displacements (i.e., uplift of the structure) are considered negative. In loading phases 1, 3, 5 and 7, measurements were not performed by BBA-TI and BBA-UAV methods due to limited time to perform measurements. Also, with RTS, displacements were not measured in points 1 and 9 (points near the beam support due to the inability to stabilize the reflector at the specified points).

These tables serve as the basis for the subsequent analysis. In the following sections, the measurement process and results for each method are presented in more detail, including the comparison with reference LVDT values and the evaluation of measurement accuracy, repeatability, and practical applicability under load testing conditions. When analyzing the difference between the displacements measured by other methods and the reference LVDT, a negative difference indicates that the measured displacement is smaller than the LVDT value, while a positive difference indicates a greater measured displacement compared to the LVDT reference.

3.1. RTS Results

Before each loading phase, the RTS coordinates were determined in relation to measurements toward two reference points (R1 and R2). As stated in Section 2.2, the unknown RTS station coordinates with the estimated standard deviation of the coordinates can be calculated using a least squares adjustment.

Since the reference points R1 and R2, as well as the RTS station, were set in the stable area, differences in RTS coordinates obtained using the free station method give insight into the achievable precision of RTS measurements. The maximum difference in RTS coordinates in eight loading phases was 0.2 mm, 0.3 mm and 0.3 mm along the X-, Y-, and Z-coordinate axes with corresponding standard deviations of 0.09 mm, 0.12 mm, and 0.09 mm. The results indicate that during the measurement process, there was no movement of RTS or orientation points and that high precision was achieved in determining the coordinates of the RTS station.

After the RTS coordinates are calculated, the coordinates of the monitoring points are determined by the polar method. Measurements were carried out by an automated procedure, where RTS independently (without the operator) searches for a prism and aims precisely at its center. Coordinates of monitoring points were determined by performing two-face measurements of the RTS to eliminate systematic errors of the RTS. Automated measurements of monitoring points by RTS lasted less than two minutes.

Comparison of RTS Displacement with Reference Values

The displacements of points for each loading phase were determined as the difference in the height of the measurement point in the current phase compared to the initial phase (before the beam was loaded). Vertical displacements determined using RTS during the loading and unloading phases of the beam, as well as the reference displacements measured using the LVDT sensor, are shown in Figure 12. Displacements by the LVDT sensor were measured at four points identical to those used in the RTS measurements (points 2, 3, 4 and 5).

Displacements determined by RTS were compared with the reference LVDT measurements at points 2–5 for all eight loading phases. The average difference for 32 measured displacements was −0.45 mm, with the differences ranging from −1.35 mm to 0.55 mm (a total range of 1.90 mm). The standard deviation of the displacement differences determined by RTS was 0.51 mm, and the RMSE was 0.67 mm, indicating high accuracy of displacement determination using the RTS instrument. Additionally, to address the expected symmetric behavior of the beam with respect to its mid-span (point 5), displacements measured at symmetric locations (points 2–8, 3–7, and 4–6) were compared. Although no independent reference measurements were available on this half of the beam, the differences between these paired points (24 displacement values in total) showed comparable accuracy, with an average deviation of −0.55 mm and a standard deviation of 0.27 mm. These results further support the reliability of the RTS measurements and the assumption of nearly symmetric beam deflection.

Figure 13 shows statistics of differences between displacements measured by RTS and referent values for each loading phase. Based on Figure 13, it can also be observed that the standard deviations, as well as the minimum and maximum values, remain at a similar level across all loading phases, indicating consistent measurement performance throughout the entire test.

3.2. HTSI Results

Within this method, displacement measurement was carried out from a single camera position located 22 m from the beam. Displacements were determined based on a single image that included both ground control points and check points signalized on the beam.

For each loading phase, a set of 10 images was taken and used to determine the coordinates of 18 photogrammetric signals. The final coordinates of each signal for a given phase were calculated as the arithmetic mean of the 10 image-based observations. Signal centers were identified using an algorithm based on edge detection, which extracted the coordinates (x, y) and radii of the circle centers in pixels.

The precision of the coordinate determination process was evaluated using the standard deviation of the 10 coordinate values from the same loading phase. The results are presented in

Table 4. Standard deviations of point coordinates based on 10 measurements (px).

Load Phase	No. of Points	Min. St. Dev.	Max. St. Dev.	Avg. St. Dev.	Load Phase	No. of Points	Min. St. Dev.	Max. St. Dev.	Avg. St. Dev.
Phase 1	18	0.2	0.5	0.2	Phase 6	18	0.2	0.5	0.2
Phase 2	18	0.2	0.5	0.2	Phase 7	18	0.2	05	0.2
Phase 3	18	0.1	0.6	0.2	Phase 8	18	0.1	0.6	0.2
Phase 4	18	0.2	0.6	0.3	Phase 9	18	0.1	0.5	0.2
Phase 5	18	0.1	0.6	0.2

.

The standard deviations of the point coordinates ranged from a minimum of 0.0 px to a maximum of 0.6 px, with an average of 0.19 px or 1.05 mm when converted to millimeters using an average scale factor.

To ensure consistency in the coordinate system across all loading phases, the homography transformation procedure described in the previous section was applied after determining the average signal centers in pixels for each phase. The transformation was performed using the ground control points (points 10, 11, 17 and 18), for which the corresponding coordinates were known in both the image plane and metric reference frame used for the robotic total station measurements.

This correction compensates for potential shifts or rotations of the camera system over time, as the test lasted several hours and small mechanical instabilities could not be excluded. Given that all signalized points were located in a single plane, the homography transformation could be validly applied to ensure geometric consistency between phases.

After the transformation, residual errors were computed at the ground control points and check points to assess the accuracy of the transformation in both coordinate directions (X and Y). The residuals for control points ranged from −0.1 mm to 0.1 mm, with an average residual of 0 mm in both coordinate directions, confirming that the applied transformation maintains a high level of internal consistency. The average residual for check points was 2.8 mm. The final vertical displacements of the points on the beam were then obtained as the difference between the transformed coordinates in successive loading phases.

Comparison of HTSI Displacement with Reference Values

The point displacements for each loading phase were determined as the difference in the (Y) vertical coordinate of the signal center between the current phase and the initial (pre-load) phase. Vertical displacements measured using the camera during both loading and unloading phases, along with the reference LVDT sensor measurements, are shown in Figure 14. Displacements by the LVDT sensor were measured at six points identical to those from the HTSI measurement method (points 1–5 and 9).

Displacements derived from HTSI measurements were compared to the reference LVDT measurements at six measurement points for all eight loading phases. The mean difference for 48 measured displacements was 0.34 mm, with differences ranging from −1.93 mm to 3.02 mm (total range of 4.95 mm). The standard deviation of displacement differences was 1.18 mm, with an RMSE of 1.21 mm. As in the previous section, displacements at symmetric points with respect to the mid-span were also compared. The deviation for these paired points consistently exhibits the same sign, with an average value of −1.45 mm. However, since this is within the accuracy level of the HTSI method, it is difficult to draw conclusions on beam behavior based solely on these differences.

Although the results were less accurate than RTS measurements, they still demonstrate acceptable accuracy for a method using a single camera position that captures the entire beam under load testing. Figure 15 shows statistics of differences between displacements measured by HTSI and referent values for each loading phase.

Based on Figure 15, it can be observed that the standard deviations and the ranges between minimum and maximum values remain relatively consistent across all loading phases. Moreover, the average displacement deviations in each phase are generally smaller than the method’s precision, as indicated by the standard deviation, confirming stable and reliable measurement performance throughout the tests.

3.3. BBA-TI Results

In the second approach, as described in Section 2.3.2, a linear photo acquisition of the beam was performed from 26 different camera positions during each test phase. The positions were arranged at 1.4 m along a line parallel to the beam at a 10 m distance.

The detection of identical points on all images and stereo matching was fully automated using Structure from Motion (SfM) and Multi-View Stereo (MVS) techniques. To maximize measurement accuracy and reliability, a high image overlap was used, with each point appearing in an average of five images. The number of overlapping images along the beam is shown in Figure 16.

The entire strip of 26 images was processed with a camera that was pre-calibrated using an on-the-job calibration. This camera calibration was further improved in the Bundle Block Adjustment (BBA) with self-calibration.

The camera was precalibrated using all markers on the beam as full control points by the self-calibration procedure of Agisoft Metashape. It was carried out on the unloaded beam, just before it was loaded. In that way, the distance from the camera to the object was kept the same during the calibration and imaging, to avoid refocusing the camera lens after calibration. The precalibration was needed because, for the orientation of the strip during the imaging of the loading phases of the beam, only four points at both ends of the beam were used, which generally is a poor distribution of GCPs for the orientation of a long strip and even more for the self-calibration and determination of additional distortion parameters. Only the points at the end of the beam were static during the whole procedure of the loading and the unloading phase. All others were moving according to the beam deformation. The lens distortion cannot be neglected because it takes on big values, especially the radial one (Figure 17). Figure 18 shows the residuals after lens calibration.

In this special case, with an unfavorable distribution of GCPs, the precalibration contributes to yielding reliable results.

Points 10, 11, 17 and 18 (Figure 7) were used as ground control points, and they influenced the orientation results, while points 1–9, 12, 13 and 16 were used as check points (monitoring points). Points 14 and 15 were excluded from the accuracy analysis due to damage during loading. The overall reprojection error after bundle adjustment was 0.244 px. The mean reprojection error on control points was 0.31 px and on check points was 0.18 px. The deviations at the control and check points are shown as error ellipses in Figure 19. The color scale on the right shows the errors in the depth measurement.

Figure 19 shows that longitudinal deviations along the beam are greater than vertical ones. Suboptimal camera network (long strip without closure or convergence), lens properties, calibration model limitations, object geometry, and potential object motion during image acquisition are all likely contributing factors to the deformation. With tilted start/end images for better intersection angles and image overlap, the camera network could have been more robust. Accuracy could be further improved by using two image strips that are slightly convergent, but doing so could run the risk of deformation of unstable objects under load. Notably, point 5, located in the center of the beam, shows the largest deviation, being farthest from both left and right ground control points. To improve measurement accuracy, adding a ground control point at the center of the beam is recommended.

The achieved accuracy for four orientation points and 12 check points in load phase 1 is presented in

Table 5. Accuracy of ground control and check points (mm) for BBA-TI method.

Type	Count	X Error	Y Error	Z Error	XY Error	Total
Ground control points	4	0.52	0.11	0.26	0.53	0.59
Check points	12	2.18	0.69	0.60	2.29	2.37

. Coordinate axes correspond to those shown in Figure 6.

Although the RMSE along the X-axis at the check points is relatively large (2.2 mm), this does not affect the measurement accuracy significantly, as beam deformation is monitored in the Z-axis direction, where the RMSE is only 0.6 mm. This makes the method competitive for tracking vertical beam displacements.

Comparison of BBA-TI Displacement with Reference Values

Vertical displacements measured using this approach during the loading and unloading phases, along with the reference LVDT displacements, are shown in Figure 20. LVDT displacements were determined at six identical points to those from the BBA-TI measurement method (points 1–5 and 9).

The displacements determined using the BBA-TI method were compared with reference LVDT measurements at six points in four loading phases (phases 2, 4, 6 and 8). The average difference for 24 displacements was −0.1 mm, with differences ranging from −1.85 mm to 1.53 mm (a total range of 3.38 mm). The standard deviation was 0.9 mm with an RMSE of 0.89 mm. Similar results were obtained based on a comparison of the precision of symmetrical points on the beam, where the average deviation for 12 displacement values was −0.6 mm with a standard deviation of 0.4 mm.

Figure 21 shows statistics of differences between displacements measured using the BBA-TI method and referent values for each loading phase. From the Figure, it is noticeable that, unlike the other loading phases, phase 4 exhibits a notably higher standard deviation and a wider range between the minimum and maximum displacement differences. This may be related to the less favorable imaging conditions of measuring the long and narrow beam in general and its impact on the results of phase 4.

Compared to HTSI results, this method provided more accurate results, albeit with a longer measurement process. Measuring beam displacements with a sequence of overlapping images allows better control of displacement accuracy by adjusting factors such as image scale, overlap (in this case 80%) and camera baseline (distance between camera positions). These parameters can be set when planning the survey, taking into account theoretical accuracy. However, this method assumes that the object remains completely static during image acquisition, which is a major limitation when tracking dynamic events.

Furthermore, to achieve high accuracy, on-the-job camera calibration is required, where all coordinates of the marked points must be determined while the beam is still unloaded, making it difficult to apply in real-world scenarios. This could be avoided by using pre-calibrated metric cameras that keep their interior orientation parameters stable during the entire acquisition process.

3.4. BBA-UAV Results

To analyze the applicability of drone surveys for determining beam displacements, a UAV survey using a DJI P4 Pro drone was conducted in a linear setup, alongside terrestrial imaging with a NIKON D750 camera configured in the same way. A total of 55 UAV positions were used for image acquisition. The distance between the UAV positions and the beam was about 10 m, while the adjacent positions along the line were about 0.6 m apart. The UAV camera was not aligned perpendicular to the beam, but the beam was captured from a height of about 1.5 m to 2.5 m. The drone was manually operated, and the camera was also manually triggered, resulting in an uneven image geometry along the beam. However, the large number of images taken along the beam made it possible to capture every point on the beam in at least eight images. The spatial resolution achieved on the beam was 2.7 mm/pixel. The number of overlapping images along the entire beam is shown in Figure 22.

The lower spatial resolution (GSD), the irregular image geometry, the lower quality of the camera and the unstable camera positions during the exposure certainly affect the accuracy of the photogrammetric measurement compared to terrestrial linear image acquisition with the NIKON D750. Excessively high overlap (90–95%) is not necessary to improve results and might even worsen deformation. The number of images in linear strips directly affects the likelihood of camera network deformation.

The deviations at the control and check points are shown as error ellipses in Figure 23. The color scale on the right shows the errors in the depth measurement.

The achieved accuracy at four ground control points and 12 check points in phase 9 is presented in

Table 6. Accuracy of ground control and check points (mm) for BBA-UAV method.

Type	Count	X Error	Y Error	Z Error	XY Error	Total
Ground control points	4	0.13	0.43	0.37	0.45	0.58
Check points	12	2.81	2.01	0.45	3.46	3.49

.

The table shows data for a phase in which there was an exceptionally large overlap between the images. Due to the manual operation of the drone, the differences in the geometry of the image acquisitions from phase to phase are very large, and the average measurement accuracy of the drone images is significantly worse compared to terrestrial images.

Although drone imaging produced the least accurate results, in situations where the object is difficult to access for measurement (e.g., bridges, viaducts, etc.), the use of drones may be a decisive factor in solving such tasks. In such cases, it is recommended to employ larger and more stable drones with higher-quality cameras and to use precise drone navigation methods to accurately position the camera during imaging in order to achieve a more regular imaging geometry.

Comparison of BBA-UAV Displacement with Reference Values

Vertical displacements measured by UAV during the loading and unloading phases of the beam, along with the reference LVDT displacements, are shown in Figure 24. LVDT displacements were determined at six identical points to those from the BBA-UAV measurement method (points 1–5 and 9).

Displacements obtained from the BBA-UAV were compared to reference LVDT measurements at six measurement points across four loading phases (phases 2, 4, 6, and 8). The average difference across 24 measured displacements was 0.06 mm, with differences ranging from −2.90 mm to 2.70 mm (a total range of 5.60 mm). The standard deviation of displacement differences determined from UAV images was 1.52 mm, with an RMSE of 1.49 mm. This method yielded the largest average deviations of symmetric points (−1.8 mm) compared to the previous techniques, but as they are at the accuracy level of the method, no reliable conclusions regarding symmetric or asymmetric beam behavior can be made.

Figure 25 shows statistics of differences between displacements measured using the BBA-UAV method and reference values for each loading phase. Phase 2 results exhibit a noticeable deviation in the average difference compared to the remaining three phases, while the ranges between the minimum and maximum values remain comparable. This indicates that, besides the deformation of the image strip, the data collected during this phase were less consistent, which in turn affected the processing and overall accuracy of the results. Once again, this highlights the strong influence of data acquisition conditions on the precision of photogrammetric measurements.

Compared to the results from BBA-TI (Section 3.3), these are less accurate; however, when compared to HTSI results, the UAV results are nearly on the same level. This is noteworthy considering the UAV camera has significantly lower measurement capabilities than the NIKON D800E used for stationary imaging. The lower quality of the UAV camera was compensated for by the denser arrangement of overlapping images in the sequence.

4. Discussion

The comparison of displacement results obtained using four different measurement methods with reference values from LVDT sensors revealed notable differences in precision, reliability, and overall suitability for various applications. Statistical analysis confirmed that all methods were capable of detecting displacements; however, they varied significantly in terms of data quality, deviation range, and potential for implementation in structural testing projects.

Table 7. Accuracy analysis of displacement measurements compared to LVDT values (mm).

Method	No. of Displacements	RMSE (mm)	Std. Dev. (mm)	Min	Max	Range	Average	Difference >1 mm (%)/>2 mm(%)	Advantages/Disadvantages
RTS	32	0.67	0.51	−1.35	0.55	1.90	−0.45	6%/0%	High precision, real-time data processing/necessary to signalize the measuring points
HTSI	48	1.21	1.18	−1.93	3.02	4.95	0.34	42%/10%	Fast measurement/lower precision (compared to RTS)
BBA-TI	24	0.89	0.90	−1.85	1.53	3.38	−0.10	29%/0%	High precision/longer measurement process and data processing
BBA-UAV	24	1.49	1.52	−2.90	2.70	5.60	0.06	58%/17%	Applicable for hard-to-reach structures/lower precision (compared to RTS)

and Figure 26 present key statistical indicators—including minimum and maximum deviations, mean values, standard deviation, and RMSE—providing a detailed quantitative assessment of each method. The RTS method exhibited the lowest RMSE and standard deviation, confirming its superior performance relative to the other methods. While photogrammetric techniques were slightly less precise, they demonstrated a high degree of accuracy and were particularly advantageous due to their flexibility and rapid data acquisition, especially in dynamic or difficult-to-access environments.

As indicated by the measurements and results analysis, each method possesses distinct advantages and limitations. The RTS method yielded the highest accuracy and reliability but involves higher costs and requires more complex technical preparation (e.g., stabilization of target points). Conversely, photogrammetric methods represent a balanced compromise between precision and ease of use, making them well-suited for field applications where structural access is limited or hazardous—such as in the case of large-span structures, tall buildings, or objects located above roads and rivers.

In addition to technical considerations, the choice of measurement method can also be influenced by the cost of equipment and software.

Table 8. Estimation of costs of equipment and software licenses for applied methods.

Method	Estimated Equipment Costs	Software	Notes
RTS	High (EUR ~50,000)	Included in the RTS instrument package	The price refers to the new model of the used RTS instrument
HTSI	Low (consumer class camera + PC, EUR 3500–7000)	Open-source or core software	Minimal costs, affordable method
BBA-TI	Low to medium (consumer class camera + PC, EUR 4000–8000 ) *	Software for SfM and MVS, e.g., Agisoft Metashape Professional, Pix4D mapper (EUR ~3500–5000 ) or open-source alternatives	Costs depend on software package and computer
BBA-UAV	Low to medium (UAV + PC, EUR 4000–8000) *	Software for SfM and MVS, e.g, Agisoft Metashape Professional, Pix4D mapper (EUR ~3500–5000) or open-source alternatives	Compared to BBA-TI, it includes additional UAV maintenance and training costs

* Software not included.

provides an overview of estimated costs for the methods applied in this research, including hardware and software license expenses. These financial aspects can play a critical role when selecting the most suitable method for a specific project, especially in budget-constrained environments.

It is worth mentioning that scale bars can be used instead of the expensive total station used in this project to determine the GCP coordinates for photogrammetric methods. This would drastically reduce the cost of the photogrammetric method equipment. For larger objects, however, the use of total stations would be necessary to reliably determine the scale definition factor.

Also, it should be emphasized that, in certain structural monitoring projects, it is necessary to track full 3D displacements rather than only vertical movements. All the applied methods—except for HTSI—are capable of determining three-dimensional coordinates, which in turn allows the calculation of displacements in the X-, Y-, and Z-directions. Although such results are not explicitly presented in this study, it should be noted that the RTS determines all point coordinates with the same level of accuracy. Similarly, both photogrammetric approaches (BBA-TI and BBA-UAV) can achieve comparable accuracy in all three directions, provided that the image acquisition geometry is carefully planned. For instance, in contrast to this experiment, where the camera’s optical axes were nearly perpendicular to the vertical side plane of the beam during imaging, using additional camera positions with tilted optical axes in relation to the vertical side plane of the beam would enhance accuracy across all spatial axes.

It should also be noted that terrestrial laser scanning (TLS) and UAV-based LiDAR are widely used in structural monitoring due to their ability to produce dense and highly accurate point clouds. Compared to close-range photogrammetry, TLS is better suited for large and complex geometries but involves higher costs, less portability, and more demanding data management. Photogrammetry, on the other hand, can achieve comparable or even higher accuracy for small-scale details at close range when using high-resolution cameras. In practice, the choice between TLS, UAV-LiDAR, and photogrammetry depends on project requirements, scale, and available resources, and these methods are often used in combination to complement each other’s strengths.

In conclusion, the choice of displacement measurement method depends on a trade-off between the required precision, available equipment, spatial and temporal constraints, and specific project demands. In practical scenarios, combining two or more methods is often necessary to achieve an optimal balance between required accuracy and applicability. Comments for each individual method are listed below.

RTS

Based on the comparison of measured displacements with reference values, the most accurate results were achieved using the RTS method. The Root Mean Square Error (RMSE), calculated from 32 displacement measurements, was 0.67 mm, while the standard deviation was 0.51 mm. Only two measurements had a deviation greater than 1 mm, with the largest deviation recorded in phase 7 at 1.35 mm (5% of the measured displacement value).

Key contributors to the high precision of the RTS method include:

• Short measurement distance,
• Accurate measurement of angles and distances,
• Clearly defined measurement points (prisms),
• Constant temperature conditions.

Additional advantages include measurement automation, real-time data processing, and immediate result transmission. It is worth noting that using an additional RTS instrument can further increase result precision and reliability but also the overall testing costs.

HTSI

The HTSI method yielded lower precision (RMSE: 1.21 mm, St. Dev.: 1.18 mm). The largest recorded deviation was 3.0 mm, with 90% of the measured displacements differing less than 2 mm from the reference values. This method stood out for its simplicity and speed—measurements, including redundancy, were completed in under two seconds.

Despite the lower achievable accuracy, this approach can be useful when measuring a structure’s response in the time domain (e.g., to determine quasi-static and dynamic displacements), assuming that the method’s precision is compatible with the expected displacement values.

BBA-TI

The BBA-TI method achieved a high level of precision (RMSE: 0.89 mm, St. Dev.: 0.90 mm) based on 24 available measurements. The maximum recorded deviation was 1.85 mm, with 71% of the data showing differences smaller than 1 mm.

Compared to the HTSI method, BBA-TI produced more accurate results; however, the measurement process lasted longer. It is also important to highlight that the precision of this approach can be influenced during the planning phase by defining parameters such as:

• Image scale,
• Overlap (in this case, 80%),
• Camera baseline (distance between camera positions).

BBA-UAV

The BBA-UAV method, although it achieved lower precision (RMSE: 1.49 mm, St. Dev.: 1.52 mm), offers advantages in practical applications where other methods are unfeasible due to physical constraints. Deviations of up to 2.90 mm were recorded, and 17% of the measured displacements differed by more than 2 mm compared to the reference values.

The UAV method is especially valuable for hard-to-reach structures such as bridges, tall buildings, or objects above roadways and rivers, where RTS or conventional cameras have limitations.

5. Conclusions

Based on the displacement measurement results obtained during the load testing of the beam, final conclusions can be drawn regarding its stability and safety under the design load. For this reason, it is crucial—already in the planning phase of the measurements—to select the optimal measurement method that ensures the required accuracy, considering the expected displacement values and specific site conditions. The selected method must align with the field conditions and the characteristics of the tested structure in order to achieve the required level of precision.

The results indicate that all the tested methods achieved highly accurate results. Additionally, photogrammetric methods have proven useful for displacement monitoring during load testing, and their application deserves further research to achieve even higher precision. Furthermore, the achievable accuracy of photogrammetric methods is directly related to the size and resolution of the sensors used, emphasizing the need for further testing of state-of-the-art cameras.

Moreover, when measuring long and narrow objects (such as this beam), where ground control points are placed only at the ends, it is essential to draw special attention to the deformation of the image strip. The deformation of the strip is mainly caused by several factors, including lens characteristics, calibration model limitations, object geometry, possible object motion during image acquisition, and a suboptimal camera network (long strip without closure or convergence). The camera network could have been more robust with tilted start/end images for improved intersection angles and image overlaps. Using two image strips that are slightly convergent could increase accuracy even more, but there is a chance that unstable objects under load will deform. It is also worth noting that the lens used in this study exhibited a noticeable drop in spatial resolution between the image center and corners, which limits the ultimate accuracy of homography-based photogrammetry. Furthermore, the relatively modest sensor resolution (40 MPx) restricted the level of detail attainable in the measurements. To minimize such effects, future studies should focus on optimizing the photogrammetric network for elongated objects, employing lenses with more uniform resolution and lower distortion, and integrating higher-resolution cameras. Additionally, the use of a more sophisticated algorithm for detecting and measuring the image coordinates of GCP would yield more accurate final results.

From the obtained results, it can be concluded that displacements smaller than 1 mm cannot be measured accurately and reliably using the tested methods. In such cases, for very small displacements measurements, it is recommended to use LVDT sensors or other precision instruments (such as a laser) which allow measurements with an accuracy of up to 0.01 mm.

Based on the conducted measurements, data processing, and analysis of achievable accuracy, general conclusions can be drawn:

RTS—Advantages: high measurement precision, automated measurement process with real-time data transmission, real-time data analysis and monitoring, ability to measure dynamic displacements with a sampling frequency up to 20 Hz, suitable for monitoring large structures without significant impact on measurement time or accuracy. Disadvantages: requires stabilization and marking of target points on structures, measures displacement at discrete points only, whereas photogrammetric methods can measure displacements of the entire structure.

Photogrammetric Methods (HTSI, BBA-TI)—Advantages: high-resolution imaging of the structure, capability to measure displacements of the entire tested structure, generation of highly accurate 3D models under load, higher precision at shorter measurement distances, greater adaptability to complex field conditions. Disadvantages: limited accuracy when applied to large structures, dependence on lighting conditions (which may change during load testing), longer data processing time compared to RTS.

UAV Photogrammetric Survey (BBA-UAV)—Advantages: rapid imaging of large areas and structures without physical contact, access to hard-to-reach or hazardous locations, ability to determine full-structure displacements, and generation of accurate 3D models of the structure under load. Disadvantages: weather dependence (e.g., wind, rain, fog), lower accuracy compared to conventional terrestrial and RTS methods, limited flight time, and legal and regulatory limitations (e.g., flight permissions). UAV use for hard-to-reach structures requires GCPs measured with sufficient accuracy (typically using a total station). Unsuitable background textures (e.g., vegetation, clouds) can impair image alignment reliability.

Each tested method has its specific strengths and limitations. The choice of method depends on the project’s requirements, the required accuracy, and the field conditions. Continued technological development will improve the applicability of all these methods in diverse scenarios. Combining multiple approaches may lead to optimal results, especially in field testing where ideal conditions for a single method are not always available.