WO2015047111A1

WO2015047111A1 - Measurement and calculation device for testing displacement of thin-walled steel elements

Info

Publication number: WO2015047111A1
Application number: PCT/PL2013/000120
Authority: WO
Inventors: Marcin DRABCZYK; Tomasz KSIĄŻEK
Original assignee: FRESH ART Sp. z o.o.
Priority date: 2013-09-25
Filing date: 2013-09-25
Publication date: 2015-04-02

Abstract

A measurement and calculation device for testing displacement of open and closed thin-walled steel elements in order to find a solution to strengthen the element that has undergone significant deformation, characterised in that it consists of interconnected PC (1) with software, measurement card (2) and optical set (3) with laser beam connected to the measurement card (2), and at least three rotation sensors (4) with accelerometer, which are placed on the tested element (5; 10).

Description

Measurement and calculation device for testing displacement of thin-walled steel elements.

The object of the invention is a measurement and calculation device for testing displacement of open and closed thin-walled steel elements in order to find a solution to strengthen the element that has undergone significant deformation.

Modern architecture is characterised by a high degree of utilisation of available materials. It employs a number of solutions to optimise the structural elements, as well as a number of computational techniques to select materials and design solutions. This allows to reduce the costs of construction, installation and materials. Structural elements are becoming more slender while simultaneously allowing to transfer greater loads and offering the possibility of spanning the structure over greater areas. This is made possible by the development of computational techniques and computing equipment, as well as the development of measurement methods which play important role in verifying the behaviour of the element in the target environment in attempts to strengthen the existing elements that are defective or do not meet some of the usability criteria (appearance, inconvenient to use, etc.) . A common difficulty that arises during the aforesaid attempts to evaluate the suitability of possible strengthening is the measurement of existing defects of the given element and the attempt to evaluate the existing values of stresses and deformations. To sum up, the structures with peculiar cross-sections are fairly widely used in Structural engineering.

The so-called thin-walled rods are an embodiment of using the characteristics and shape of material. They are characterised by a considerable lightness and can be used to form a variety of shapes. The choice of the shape is in such case essential to the improvement of the strength of the cross-section. Material that is used in such structures can range from steel to concrete.

In many constructions, thin-walled rods are used often in the form of hot rolled shapes and shapes made of cold-bent sheet metal strips. These elements can be combined with each other and with coatings to form, among others, metal frames of motor vehicles, roof structures, bridges and others. Embodiment of cross section of such an element is shown in Fig. 1.

In general, to classify an element as thin-walled the condition d/6>8-10 is checked, where "d" is the width or length of the wall and δ is its thickness. In other words, a thin-walled element is an element whose one dimension determining the cross-section (thickness) is strikingly smaller than the other one. To sum up this classification, the elements can be considered to be thin-walled in the following cases: a) Cross-section wall thickness δ is at least eight times lower than the longest path measured along the centre line - AB path in Fig. 1

b) Path z is at least eight times shorter than the length of the rod L:

1

z < - * L

(S. Piechnik [2000]. The second of these conditions is presented in the textbooks on designing steel structures - for example (J. Zmuda [1997]) in a slightly less restrictive form, namely: bl) in cases of beams under lighter load, i.e. distributed load

1 1

— * L≤ h <— * L

12 10 b2) for beams under heavy load, i.e. concentrated load 1 1

— * L≤ h≤- * L

10 8 where: h - height of the cross-section of the rod. In view of the desire to analyse the numerical examples, mostly congruent with the challenges encountered in engineering practice

In addition to the obvious advantages of such solutions, there are also issues arising out of their application. The main problem of such systems involves high degree of complexity of the calculations, as well as the verification and testing of the systems existing in construction industry. The main reason for this is the fact that thin-walled cross-sections are often prone to warping due to their unique dimensions, as mentioned above. The deformation of the element along its length is non-uniform and causes additional stresses that need to be considered during dimensioning and verification. Thin-walled rods are characterised by their vulnerability to the manner and area of applying load.

Embodiments of thin-walled cross-sections:

1) Metal structures

a) (hot) rolled profiles.

- standard I-beams

- I-beams with parallel flanges

- H-beams

- C-beams

- equal angles

- unequal Angles

b) Cold-bent profiles (Fig. 2)

2) Concrete structures

Reinforced concrete structures are generally free of thin-wall related effects, but there are exceptions to this, such as bridge-like cross section (Fig. 3): The prior art discloses methods for conducting such research by commissioning a research institute to analyse the problem. The institute carries out a series of tests using devices designed to test abstracted samples only. The tests include a variety of operations to simulate the actual behaviour of the element within the structure. The calculations are carried out on models based on the results of experimental tests and the measurements of the existing structure. This process is both complicated and, more importantly, time- consuming. It also involves a lot of effort. As of yet, no devices were found that would, within one cycle, measure displacement, calculate strength and adjust the cross-section by strengthening the element.

The proposed device greatly simplifies the verification cycle and shortens the time of testing. The device can be operated by only one person. This person must have basic understanding of the given issue, as issues of this kind are not common knowledge. The "classical" approach often required more than one person to be active in developing the study. This was due (in cases where it was necessary) to the need to prepare the samples and test them.

A measurement and calculation device for testing displacement of open and closed thin-walled steel elements in order to find a solution to strengthen the element that has undergone significant deformation, according to the invention, is characterised in that it consists of interconnected PC (1) with software, measurement card (2) and optical set (3) with laser beam connected to the measurement card (2), and at least three rotation sensors (4) with accelerometer, which are placed on the tested element (5; 10), whereas:

1) the rotation sensor (4) consists of an accelerometer (6) located on a mounting platform consisting of a holding table (7) and a telescopic part (8) that slides out of it, whereas the holding table (7) base that is in contact with the tested element (5) and the telescopic part (8) is equipped with fastening magnets (9); 2) the optical set (3) consists of:

a) base part located on a stabiliser on the tested element (5; 10) and consisting of a photoelectric sensor (25; 26) and high-definition camera (27) with red guiding spot light;

b) movable part consisting of a measurement board (30) mounted on the stabiliser in the measuring points of deflection of the tested element (5; 10).

Preferably, the telescopic part (8) consists of the first arm (12) and the second arm (13) that slides out of the first arm, on which the fastening magnet (9) is mounted.

The solution according to the invention is designed to be used by persons or institutions that carry out research or studies, construction supervision authorities, designers or construction contractors. The device is also mobile, easy to transport and has small dimensions, which is crucial for making measurement tests on an existing structure. Another preferable feature of the device according to the invention is the ease of replacement of its components subject to wear, as well as the availability of the components on the market, so that the costs of use are minimised. A significant part of the device design, in addition to measurement mechanisms, is the use of innovative software solutions that enable control and readouts of the measurement elements and carry out calculations based on the finite element method - necessary to estimate the possibility of strengthening the tested element. The software automates a number of processes that previously were performed step by step and using independent solutions, which could affect the performance and correctness of the proposed solution in the final stage of verification/study. It should be noted that the compactness as well as consistency of the solution substantially improve the performance and reliability of the test and, consequently, the proposed calculated solution to the problem. It obviates the need to carry out expensive analysis with the use of external calculation software which generates only indirect calculation data. A thin-walled cross-section beam subjected to a load either in the form of concentrated force or other force will be deflected in the plane of the applied load and subjected to torsion around its x-axis. In addition, buckling (warping) occurs. Determination of the deflection and tension at each point is not possible to be determined in this case, based on the theory of solid rods (namely: Bernoulli's principle for flat cross-sections and de Saint-Venant principle), i.e. rods whose walls have greater thickness relative to their length. Such a case requires an analysis of the angle of rotation (warping) of the thin-walled cross- section around the longitudinal axis of the rod. The analysis will not consider the instability of the cross-section walls, i.e. it will not consider the local instability, but only the deflection-torsion one (Fig. 4).

Embodiment of the design is shown in the drawing in which Fig. 5 shows the schematic structure of the device, Fig. 6 is a chart of the flow of information between the controller (PC) and the rotation angle sensor, Fig. 7-15 show the schematic structure of the rotation sensor and the stabiliser on which it is placed, Fig. 16 - suggested locations of the optical set, Fig. 17 - chart of the flow of information between the controller and the measurement board, Fig. 18-19 - schematic structure the optical set, and the stabiliser on which it is placed, Fig. 20 - graph of the readouts of the light spot from the measurement board, Fig. 21-23 - schematic structure of the measurement board and its stabiliser, Fig. 24 - view of the measurement board, right type, Fig. 25 - view of the measurement board, left type, Fig. 26 - communication chart between the algorithms in the control module of the PC, Fig. 27 - method for mounting the optical set on the tested element, Fig. 28 - flowchart of the steps to take while defining the boundary conditions, Fig. 29 - flowchart of the steps to take during reconstruction analysis, Fig. 30 - view of the stresses in the cross-section of the element relative to the plane of the thin-walled elements, Fig. 31 - flowchart of the possible modes of calculations made by the device (theoretical analysis and practical analysis based on the readout of the sensors), Fig. 32 - 35 flowcharts and modes of measurements and calculations, Fig. 36-41 - examples of practical application of the device to verify the correctness of operation. In order to determine the necessary parameters (vertical and horizontal displacement, as well as the angle of torsion of the cross-section causing buckling), a set of sensors was designed, with each sensor serving different purpose. The sensors can be divided into two groups. a) The rotary set is responsible for reading the angles of rotation of the cross-section.

b) The displacement set consisting of an optical sensor and a readout device

The sensors communicate with the control device, in this case a PC, via custom-developed software that converts the electrical impulses from the sensor to digital data for later processing within the task. Fig. 5 presents the overall schematic structure of the device. The device consists of a PC connected to the measurement card, which in turn is connected to the optical sensor with a laser beam and rotation sensors equipped with an accelerometer (in this embodiment there are four sensors) where:

1 - PC with software

2 - measurement card 3 - optical set

4 - rotation sensors (4 pieces) with accelerometer to test the torsion angle at characteristic points

5 - tested beam.

The device uses a standard PC (Fig. 5 item 1) available on the market with a USB port. In order to guarantee efficient operation, the PC should meet the following minimum requirements: - Intel core Ί5 processor (8M Cache, 2.66 GHz)

- 4GB RAM

- 200 GB hard drive

- GeForce GT630 1024MB graphics card

- Windows XP or newer

- lxUSB

The measurement card (2) connected to the PC is a versatile device that allows to automate the measurement processes at research stands in laboratories and to control the production environment in real time in industrial plants. The USB interface of the card allows it to be easily and quickly connected to any PC with a USB 2.0 port, including notebooks.

The function of the communication devices, such as the measurement card (2), is to acquire the data (capturing and sampling data, capturing signals from the environment so that they can be sent to the PC and processed as required; data acquisition includes gathering data, signals and waves, and processing them in order to achieve the desired results and information) and to control. It is characterised by complete independence from stationary systems and their physical limitations (hard to reach places, distances, etc.). The modules are powered directly via the USB port of the laptop. They do not require any additional accessories and are equipped with insulated I/O channels and ESD protection.

Specifications of an embodiment of the measurement card:

• Number of analogue inputs: 16

• Resolution: 16 bits

• Processing speed: 200k samples/s

Input ranges: ±10V, ±5V, ±2.5V, ±1.25V, ±0.625V

• Number of analogue outputs: 2 Output ranges: ±10V, ±5V, 0÷10V, 0÷5V • 8 TTL digital inputs

• 8 TTL digital outputs

• 1 counter (32 bits) USB 2.0

• Powered via USB port (+5V)

• Power consumption: max 440mA

• Dimensions: 132x80x32mm

• DLLs supplied for Windows 2000/XP/Vista

In the embodiment design of the device there are four independent rotation angle sensors (4). Each sensor is independently controlled and has an independent channel of signal sensing. The number of sensors is chosen to best interpolate the angle to the beam length in one step. The measurement quantity can be copied by selecting the appropriate option of the supplied software of the control unit. This will sum up the number of measurements obtained in "n" steps.

The rotation sensor consists of two main parts. The first part is the measurement element - accelerometer. The second part is a custom-designed system of mounting the accelerometer, used for precise determination of the angle of rotation of the cross-section.

The device is equipped in two modes of sensor operation: a) Measurement mode

In this mode of operation, the actual measurements are made for further processing.

b) Calibration mode

Basic parameters of the sensor, such as its orthogonality, can run out of adjustment as a result of its use. In this mode, the sensor can be calibrated against the reference values, using the appropriate option of the control panel. Fig. 6. shows a chart of the flow of information between the controller (PC (1)) and the rotation angle sensor.

The accelerometers are used to measure the acceleration of the object (sensor). We can distinguish both static acceleration (i.e. gravitational or magnetic forces) and dynamic acceleration (directly related to the change in velocity during movement). The measurement of linear acceleration allows to estimate the strength affecting the object (if its weight is known), as well as its speed and location.

In the embodiment design, the used three-axis ADXL 335 accelerometer provides linear acceleration values within the range +/-3.6g (lg=9.81m/s2). This configuration allows to estimate the net static acceleration in any direction, caused by gravity.

ADXL 335 accelerometer employs Micro Electro Mechanical Systems. Stationary combs mesh with comb-shaped beam, which is put out of balance by acceleration.

ADXL 335 accelerometer is a compact size (4x4x1.45 mm) analogue sensor of linear acceleration with power supply ranging from 1.8 to 3.6V and low power consumption of about 350μΑ. As stated by the manufacturer in the product sheet, the typical measurement range of the sensor is ±3.6g. This value is sufficient for the purpose, as the system is not designed for high-dynamics objects. The level of measurement linearity does not exceed 0.3%. It can be therefore assumed that the measurement results are directly proportional to the actual acceleration. Since the measured range includes both senses of the acceleration in each direction, the output data of the sensor are offset with the value corresponding to Og. As declared in the product sheet, this value for each axis is typically 1.5V. The resolution of the sensor is 300mVg.

Assuming that the only acceleration applied to the sensor is static acceleration of gravity, ADXL 335 accelerometer can be used to estimate the value of the tilt from the vertical plane. Since the measurement results of the three axes are available, the angle range of the tilt is (-π, ni). Individual acceleration on each axis can be calculated from the following equations:

Vxout — Vxoff

Sx

~ Vyoff

Sy

Vzout - Vzoff where A_x, A_Y, A_z are acceleration values expressed in g, V_Xout, V_Yout/ V_Zout stand for the voltage levels measured on the accelerometer outputs V, V_Xoff, V_Yoff, V_Zoff represent offsets for each axis, i.e. the value of voltage in volts for the acceleration equal to Og, and where S_x, S_Y, S_z is the sensitivity of each axis in Vg. Two Euler angles (roll (p) and pitch (φ)) can be thus determined using the following equation:

In addition, the angle Θ, which is the overall tilt of the XY plane from the vertical plane, is defined as:

The atan2 function is a binary version of the trigonometric arctangent function that returns the angle value from the range (-π, π).

The sensor itself was placed on a special platform that allows it to be attached to the tested element. Please note here that the sensor should be properly calibrated after mounting it on the platform as, due to technological reasons, it can deviate from orthogonal mounting. In addition, during operation, the orthogonal orientation might be skewed as a result of the wear of the element material or other unforeseen factors. For such cases, the master device provides a calibration tool. The calibration is carried out on a properly prepared surface. In this case, the surface should be perfectly horizontal and the plane perfectly vertical. Deviations from these conditions should be defined as measurement error. Fig. 7 shows the diagram of the platform for mounting the sensor to the tested element. where:

6- Accelerometer mounted on the mounting platform

7- Table holding the accelerometer to the tested element - the stationary base part

8- Movable telescopic part of the holding table

9- Fastening magnets protruding by 5 mm

10 - Tested element

The mounting platform transmits the rotation angle of the cross-section from the element onto the accelerometer. This component is designed so that the entire sensor system can be spanned over the characteristic points of the cross-section. This is crucial in case of a possible small deformation (possible buckling) of the cross-section walls. To meet the above, a table has been constructed with a retractable arm, at the end of which fastening magnets are mounted. This table is telescopic, i.e. each successive arm extends from the base arm behind it. This allows to maintain compact size and minimise the deviation of the measured values to a negligible level. Additionally, a limiter is provided in the form of stop pins which prevent the arm from extending beyond a certain level. It is relatively important to ensure maximum accuracy of the extended item. The table includes openings on two different movable parts. The first one is the base part, whereas the next one is the last movable element. Magnets were inserted into the openings and (by way of an example) attached with an adhesive. The magnets protrude 5 mm in relative to the cross-section of the table. This allows to make free space for any unevenness of the wall caused by i.e. buckling.

Parameters of an example telescopic part (table)

Table 1

The drawings in Fig. 8-12 show the cross sections and the detailed design of the table. where:

6- Accelerometer

7- Base part of the table with the fastening magnet

11- Opening for the fastening magnet

12- Movable first arm of the telescopic part (8)

13- Second arm of the telescopic part (8) with the fastening magnet. Movable part. The maximum radius of the arms is 150 mm

The drawings in Fig. 13 show the extended telescopic part where:

6- Accelerometer

7- Base part of the table with the fastening magnet

11- Opening for the fastening magnet

12- Movable first arm of the telescopic part (8)

Optical technique was used to determine the next key measurement parameter, i.e. the deflection in two planes. An important factor in this type of measurement is the knowledge of the exact location of the measurement cross- section. The optical technique used in the device includes this factor.

An optical set (3) was applied that includes a photoelectric sensor comprising an optical rangefinder with a generated red spot light and a measurement board mounted at the measurement spots. In addition, a high- definition camera was used to read the image from the measurement board for further digital processing. The optical set (3) should be installed at the location of the "fixed" boundary conditions - in other words, at the end of the element with the highest stability. It is also necessary that the optical set (2) is as close as possible to the theoretical point of support. This is necessary to obtain correct readouts. The mounting offset should be measured using the available methods and introduced as an option in the control unit software. This parameter will be considered in later analysis. Mounting the optical set in a place that is not supported should be avoided, as there is a risk of the optical set exerting load on the element. Fig. 17 shows the suggested location of the optical set (3) (black box), provided that the sub item b) is disregarded when installing the set.

Generally, the optical set can be divided into two parts:

1) The base part including

a) A photoelectric sensor with a built-in red guiding spot light.

This sensor is used to determine the distance of the measurement board mounted on the tested element; additionally, the red spot light will indicate the displacement location of the measurement board. The sensor is a component generally available on the market and is mounted on a special holder. b) High-definition camera

The purpose of the camera is to capture the measurement data from the measurement board. An integral part of the camera is a software that analyses the images to locate the point of reflection of the light beam from the optical sensor on the measurement board. 2) The movable part including

a) The measurement board

The measurement board is mounted in measuring points of deflection of the tested element. The white board includes a grid (horizontal and vertical lines) with a millimetre scale.

The optical set operates as follows. After properly installing the components of the set (the base part), several measuring points are selected to achieve best results. It is suggested to include 4-5 measuring points to accurately recreate the course of the deflection lines. The measurement board is mounted at these places. The device, via the camera located in the base part, makes a readout from the measurement board by measuring the location of the light spot on the scale on the board. The data is then transmitted to the PC for processing. Fig. 17 presents the chart of the flow of information between the controller and the measurement board.

The base part of the optical set (3) is a set of two components mounted on one stabiliser. To assure proper cooperation between the two elements serving different purposes, the stabiliser was designed according to the following principles and guidelines:

- the stabiliser should be a stable system that minimises any vibrations transmitted due to the behaviour of the structure;

- the stabiliser should allow the entire part of the sensor to be horizontally extended. To this end, a telescopic system was used with a friction lock;

- the stabiliser should allow to measure the distance from the start of the system to the edge of the cross-section element. It is important for the measurement of horizontal displacement. To ensure that these criteria are met, the chosen solution is similar to the measurement of the distance between the measurement board and the optical set. An optical sensor with a laser beam was used. The sensor measures the distance between the laser beam and the flat wall of the cross-section. This allows to determine the location of mounting the system, which should coincide with the centre of gravity crucial in measuring deflection. WTB 140-P330 sensor was selected

Table 2

- the stabiliser should also ensure that the system is levelled to guarantee proper readouts. This is necessary to determine vertical deflection, assuming that the neutral axis coincides with the axis of the centre of gravity of the cross-section. The arm of the telescopic system is mounted on a hinge with an inclination angle lock. In addition, to precisely align the research instruments, a swinging system was used consisting of three screws that adjust the position of the base. The position of the base is adjusted by tightening and loosening the three screws. This is a system similar to that used in levellers.

Fig. 18 shows a schematic structure of the stabiliser - side view where:

14- Mounting table - attachment with magnets to the wall of the thin-walled profile. Provides stability.

15- Openings for fastening magnets. The magnets protrude 5 mm below the table, allowing freedom of movement.

16- Telescopic system. Base part

17- Screw locking the movement of the telescope in the base part 18- Screw adjusting the position of the metric part (the board)

19- Sliding element of the telescope

20- Base of the swinging system

21- Socket of the screw adjusting the position of the metric part

22- Inspection opening

23- Mounting plate

24- Reinforcement brace

25- Photoelectric sensor with laser. Part fastened to the mounting plate

26- Photoelectric sensor with laser pointing at the centre of gravity of the cross-section. Part fastened to the mounting plate

27- Camera

Fig. 19 shows a schematic structure of the stabiliser - top view where:

16- Telescopic system. Base part

17- Screw locking the movement of the telescope in the base part

18- Screw adjusting the position of the metric part (the board)

19- Sliding element of the telescope

20- Base of the swinging system

21- Socket of the screw adjusting the position of the metric part

22- Inspection opening

23- Mounting plate

24- Reinforcement brace

25- Photoelectric sensor with laser. Part fastened to the mounting plate

27- Camera The type of photoelectric sensor included in the optical set (3) and used in the device according to the invention to determine the exact distance between the optical set and the tested cross-section may be the type of photoelectric sensors generally available on the market, whose parameters correspond to those in the documentation of a miniature reflective sensor with background suppression WL-140 Photoelectric Reflex Switch, Red Light - DC by SENSICK.

The above type of sensor has been selected due to its satisfactory performance. The operating distance of 6.5 m is established as the maximum distance of the recently tested cross-section. Sensors with better parameters can be used to improve the operating parameters and thus increase the reading distance.

Sensor parameters:

Table 3

The sensor is connected with the measurement card. The control unit software is used to determine the distance by averaging the data transmitted from the measurement card.

The camera (27) included in the optical set (3) and used to capture the position of the light spot on the measurement board, is a board camera generally available on the market. This camera has been chosen because of its small size and weight, as well as its ability to precisely analyse the captured image.

The parameters of the camera can be identical or similar to the parameters of RHPC-2005 board camera by RHP International.

Parameters of the camera: Table 4

The camera (27) is connected directly to the PC (1). The software analyses the transmitted image in relation to the measurement board. During the measurement, the camera captures images which are saved as image files. The software processes the images in search of the pattern on the measurement board. Once the patter is found, a special algorithm for pattern recreates the board in the memory and proceeds to localise the light spot generated by the optical sensor. Two following fundamental physical problems occurs during the analysis: a) The light spot has different focus depending on the distance from the source.

To overcome this issue, the algorithm localises the centre of the area that has been chosen as the light spot area. This is not a difficult task, considering that a red spot is being localised on white background.

b) The problem of vibration of the light beam depending on the temperature, humidity and air movement.

To overcome this issue, a series of images is analysed depending on the assumed precision level. The images are processed and categorised according to sub-item "a)" and then the average is calculated. If the average of two considered time periods is lower than the assumed acceptable standard, the measurement result is accepted.

Fig. 20, as the readout of the light spot from the measurement board, shows the possible position of the light spot obtained after averaging the position of the spot on the measurement board. The vertical axis is the axis of the measured position, while the horizontal axis is the time of measurement. The measurement period was chosen equal to 100 units. After this period, the result is averaged as shown by the black line. Another averaged readout is carried after additional 100 units. After this period, the results are compared with the standard.

\w_T(n) - w_T(n+1) \≤ norm

where:

W_T(n> is the average value from the given time period (100 units in this example)

W_T (n+i) is the average value from the subsequent time period (100 units in this example)

norm - allowed deviation of the measurements between the time periods.

The measurement board (30) is an integral part of the optical set (3). The measurement board (30) is used to read the measurements. The measurement board is a flat (board-like) item that is mounted to the tested beam on a moving stabiliser. The component parts of the measurement board: a) Stabiliser - responsible for maintaining level and for the measurement of the distance of metric part from the edge of the tested cross-section. b) The metric part, also referred to as the board, is used to read the measured values from the external components of the main unit. An important parameter of the entire system is its weight. Weight is essential in case of elements with small cross-sections, due to the fact that the load of the measurement board may affect the measurement results of the deflection of tested elements. The weight of the entire system and the values of the forces acting on the tested element are shown in Table 5

Table 5

The stabiliser of the measurement board is a component for mounting the board on a tested element. The light beam is emitted from the transmitting element (the photoelectric sensor in this case). To enable reading of the position of the light spot on the metric part, it should be possible to fully adjust the board with the possibility to read the level of adjustment and send the readings to the control unit. To ensure that the above are met, the stabiliser should meet the following conditions:

- the stabiliser should allow the measuring part of the board to be horizontally extended. A telescopic system was used with a lock.

- the stabiliser should allow to measure the distance from the start of the board to the edge of the cross-section element. It is important for the measurement of horizontal displacement. To ensure that these criteria are met, the chosen solution is similar to the measurement of the distance between the measurement board and the optical set. An optical sensor with a laser beam was used. The sensor measures the deflection of the board from the edge of the element to which the set is attached. In addition, it determines the central point of the set, i.e. zero axis. The theoretical deflections assumed in the model are calculated for the so-called centre of gravity axis of the cross-section. The board (or more accurately the "neutral zero axis") should coincide with the axis of the centre of gravity of the tested cross-section - it is important for vertical deflections. In this case WTB 140-P330 sensor was selected

Table 6

- the stabiliser should ensure that the board is levelled to guarantee proper readouts. This is necessary to determine vertical deflection, assuming that the neutral axis of the board coincides with the axis of the centre of gravity of the cross-section. The arm of the telescopic system is mounted on a hinge with an inclination angle lock. In addition, to precisely align the position of the board, a swinging system was used consisting of three adjustment screws.

Fig. 21 shows a schematic structure of the measurement board stabiliser - rear view where:

16- Telescopic system. Base part

19- Sliding element of the telescope

20- Base of the swinging system

21- Socket of the screw adjusting the position of the metric part

22- Inspection opening

28- Mounting plate

29- Photoelectric sensor with laser pointing at the centre of gravity of the cross-section.

30- Metric part with the scale and pattern.

Fig. 22 shows a schematic structure of the measurement board stabiliser - front view where:

16- Telescopic system. Base part

17- Screw locking the movement of the telescope in the base part

18- Screw adjusting the position of the metric part (the board)

19- Sliding element of the telescope

20- Base of the swinging system

21- Socket of the screw adjusting the position of the metric part

30- Metric part with the scale and pattern.

Fig. 23 shows a schematic structure of the stabiliser - top view where:

20- Base of the swinging system

28- Mounting plate

29- Photoelectric sensor with laser pointing at the centre of gravity of the cross-section. Part fastened to the mounting plate - Metric part with the scale and pattern.

The board (30) has a vertical and horizontal scale with a measuring edge located on the side edges of the board. The vertical and horizontal scales form perpendicular and parallel lines that resemble checked surface (similar to grid paper). The distance between the lines is 5 mm x 5 mm. This ensures satisfactory accuracy of the measurements, examined to be at the level of 0.1-0.2 mm, depending on the distance between the optical part and the measurement board. The distance between the lines is inversely proportional to the accuracy of measurement. It is due to the size of the light spot generated by the laser. If laser with better parameters is placed in the base element, the accuracy will be greater. In addition, the outer top corner includes a pattern (marker) that aligns the digital images captured by the camera.

The shape of the pattern is typical: a circle with four separated areas. Two areas that are located opposite each other are black, provided that the black bottom quarter area is located on the inside - in other words, on the side of the tested cross-section.

Two types of boards were used: left board (Fig. 24) and right board (Fig.

25). where:

30- Board with vertical lines

31- Marker

32- Horizontal lines

33- eutral "zero" axis

Depending on the side of measurement, the appropriate type of board is used.

The components of the device can be divided into two groups: the set of sensors and the control unit. The control unit (in this case a PC) carries out a series of complex analyses, processing operations and calculations. These can be divided into the following groups: a) Analysis and processing of external signals from the sensors and the camera.

b) Parameterisation, calibration of the components of each sensor. This process requires precision.

c) Analysis of the input parameters and calculation in order to obtain the desired effect.

This three-stage process involves the usage of a set of software algorithms that are characterised by high level of innovation and efficiency. These solutions have not been found in the engineering software available on the market.

A diagram of the communication between the data algorithms employed in the control unit, together with the classification, is given in Fig. 26 as an information flow chart.

Analysis and processing of external signals

The output signal from the sensor indicates the value of the voltage on each component. The measurement card (2) is responsible for the management, clocking and sequencing readouts. The measurement card (2) transmits the relevant information to the control unit in the PC (1). The control unit subsequently processes and digitises the signals. The information provided by the measurement card (2) to the control unit is the voltage on the given transmission channel. With this information, it is possible to convert the voltage values to relevant characteristics, such as the angles of rotation (tilt) for the accelerometer and the distances transmitted from the photovoltaic rangefinders. The main issue connected with analysing the signal involves noise. The noise is caused both by the characteristics of the measured variable, as well as the unreliability of the measuring equipment. Each readout of a physical characteristic provides a slightly different value. To alleviate this, the readouts are averaged.

In digital signal processing, the averaging often involves totalling the sequence of signal samples versus time and dividing the obtained value by the number of the samples. In mathematical terms, the average of N samples of x(n) sequence is referred to as x_ave is expressed as:

The key parameter in averaging is variance σ² defined as:

The second used parameter is standard deviation, defined as the positive square root of variance:

The applied averaging method was incoherent averaging. The process of incoherent averaging (also known as RMS, postdetection or video averaging) is the averaging of signal samples where no sample timing constraints are used; that is, signal measurement time intervals are not synchronized in any way with the phase of the signal being measured.

Another independent source of data necessary in the analysis is the data received from the camera. In this case, video signal is fed to the control unit. The fed video signal needs to be stored. During the analysis, the representation of the image is stored in the computer memory.

Image analysis involves extracting the part which is important for the process from the total information that reaches the detector. This is carried out by eliminating a vast part of the information. Out of several thousands (or even millions) of bytes that, in typical conditions, constitute each image, only a few dozen or a few hundred bytes are left that represent the values extracted during the analysis of the parameters of the objects that make up the image. Image segmentation is the process that connects the layer of image processing with software analyses of the respective objects. This process involves dividing the image into the sections corresponding to the objects visible in the image. At the same time, the objects are indexed, i.e. the selected pixels of the objects are assigned identifiers that indicate which object is represented by the given pixel. Segmentation is therefore an image processing technique that allows to distinguish the areas of the image that fulfil the criterion of homogeneity. In the case of the aforesaid analysis, this criterion is the colour of the area. Segmentation by incremental increase of the area and segmentation by edge detection (these are the used ones) are used as a preparatory stage in object recognition.

Segmentation by edge detection is used to detect horizontal and vertical lines and thus to recreate the grid formed by these lines on the measurement board. This method searches for the edges between areas. This is accomplished by applying gradient operator and, subsequently, applying threshold to the gradient. In the next step, the elements of the image that have been identified as edges are separated as independent objects. The applied technique involves an analysis of the differences between two groups of elements, in a manner similar to linear high-pass filters. More than one filter needs to be applied in order to take into account the differences in the orientation of the edges.

Segmentation by incremental increase of the area is used to detect the pattern to orientate the observed graphical objects. This method searches group of elements with similar level of brightness. The simplest form of this method involves starting from one element and checking whether the neighbouring elements have similar brightness. If so, they are grouped together as one area. This way, areas are created that expand from individual elements of the image. In the next step, each area is tested for uniformity. If the area is not uniform, it is once again divided into smaller parts. One uniformity criterion is based on comparing the maximum difference between the value of the image element L(m,n) and the average value for the area. Mean for the area R is calculated:

Then the following condition is checked: max_mfneR \L(m, n)— S\ < T

The area is uniform if the above condition is satisfied for the chosen threshold T. The choice of this threshold is based on the following fact: the probability that the brightness of the element L(m,n) diverges from the average S by more than a value x is expressed by the integral:

where: σ - standard deviation of the noise

After segmentation, the objects are indexed, i.e. all pixels of the objects are assigned with identifiers that indicate which object is represented by the given pixel. This is done by assigning special values to the pixels, which corresponds to filling the areas representing the discerned objects with the shades of grey introduced for this purpose. Indexation allows to carry out all measurements on individual objects, as the software is given the boundaries of each analysed object.

Measuring is the final stage of image analysis which is subject to two categories of parameters: a) Local parameters that describe an averaged element of the image. In the case of the described problem, it is the average size/diameter of the object.

b) Global parameters the describe the characteristics of a group of objects or other elements of the image. In the case of the described problem, these is the length of the lines per unit of area of the image. After completing the measurements of the image, the values that describe the objects and that are further processed to determine their characteristic features are obtained. The main purpose of the analysis of the shapes is to determine such features of the objects that adequately describe the shapes of sought objects. Identical values are taken for objects with identical shapes.

Conversion the defined data to a numerical model

Defining basic input parameters is not an automated activity. This concerns the following input parameters a) Mounting distance of the optical assembly on the beam.

This value is manually entered by the user. In practice, it is not possible to place this type of sensor on the theoretical axis of the tested element. For this purpose, the optical set (3) is mounted in a certain distance as close as possible from that axis. This distance must be taken into account when extrapolating the displacement values at the point of divergence of the theoretical central axes of the elements. The distance should be measured from the point of the laser beam indicated by the measuring element (optical set). The system automatically takes into account this correction in later calculations and considers any difference in the measurement. This situation is shown in Fig. 29 where:

3- Optical set

10- Tested horizontal element

34- Vertical element

35- Central axis of the tested vertical element

36- Central axis of the tested horizontal element

37- Axis defined by optical set laser

d- Measurement distance

b) Parameters of the cross-section. The parameters of the cross-section are manually entered using the cross-section definition module. Each new inserted element may be saved in the device for storage. The device is equipped with typical cross- sections, together with their defined parameters, as well as parameterised cross-sections, i.e. cross-sections of a predetermined shape and with the possibility to define the thickness of each edge. This tool, after defining the parameters, calculates the geometric and fragmentary characteristics and the deflection centre - the point at which the applied shearing force does not cause twisting moment. By way of an example, the following are some of the formulas for these characteristics: - geometric characteristics

• Static moment relative to the Y and Z axes

Sy = j z(s)6(s)ds S_z = j y(s)6(s)ds

Centrifugal moment

j y(s)z(s)S(s)ds

Moment of inertia relative to the Y and Z axes

Iy = j z²(s S(s)ds

j y² (s) S(s)ds

- deflection centre

'ωΒγ

+ Z_B fragmentary characteristics

• Fragmentary static moment

Sco = j w(s)S(s)di • Fragmentary centrifugal moments

z = j ( (s)z(s)6(s)ds

• Fragmentary moment of inertia

Ι_ω = j oo²(s)6(s)ds where:

6(s) - wall thickness

The mentioned characteristics are important in the selection of the location of not only the optical set but also the measurement table (30), which should be, as far as the position on the cross section is concerned, located in exactly the same place as the optical set.

Boundary conditions.

Quoting Wikipedia, boundary conditions can be defined as the task of determining, among the functions of a given class (for example satisfying a given differential equation) defined in the considered area, those that meet the additional conditions on the boundaries of the area. Such conditions are referred to as the boundary conditions and are imposed on the values of a function and its derivatives in more than one point of the area.

In order to recreate the actual behaviour of the tested element by the device according to the invention, the boundary conditions with relation to the element shall be considered. For this purpose, the device provides a set of common, variously formed nodes connecting the tested elements with the adjoining elements. Additionally, third parties can define their own types of connections that can be used in modelling the element in the device. Please note that proper selection of nodes is a key prerequisite for obtaining correct results at the end of the process. Fig. 28 shows the workflow chart.

Boundary conditions, in addition to geometric type, are assigned the measured local rotation of a given node. If it is difficult or impossible to carry out the above measurements, the software allows to estimate by extrapolation the rotation and displacement of the boundary places of the element. In addition, the rigidity of the nodes is defined by introducing the length of the elements adjoining that node. This length is converted to the vulnerability of the fastening of the given node fragment. The so-called replacement rigidity of the element is introduced, which depends on the geometry of the adjoining element. This is solved by considering support and boundary conditions in the structure rigidity matrix. The support conditions are established in the form of supporting elements with the following rigidity matrix:

0

[k] = 0 k_x C

0 0 k where:

K₂ - linear rigidity along the Z axis

K_x - torsional rigidity around the X axis

K_y - torsional rigidity around the Y axis

• linear rigidity along Z axis

EA

• torsional rigidity around the X and Y axes for an element fastened constrained on both sides

K - ^4EJy

y^{~ ~}h ~

• torsional rigidity around the X and Y axes for an element fastened constrained on one side where:

E - modulus of elasticity of element material

A - area

Jx and y - moment of inertia of the cross-section respective to the X and Y axis

h - the height of the element

Recreation of the numerical model using the data from the sensors.

An important issue in the correct recreation of the stresses and deformations in the tested element is the correct representation of the sensor data. The principle applicable for the interpolation is as follows: the more measurement points the more accurate representation of the displacement graph. This is due to the interpolation method. After a series of tests, it was found that the sufficient accuracy is achieved for 4 to 6 measurement points, depending on the load. The chosen method of interpolation was spline interpolation. Fig. 31 shows a flowchart of a recreation analysis.

Spline functions are an implementation of smooth local interpolation by a third degree polynomial with a smooth spline of the respective local polynomials. Since an N-l degree polynomial is unambiguously defined by N equations, the subsequent equations can be obtained by imposing the continuity of the first and second derivative at the point of spline of the polynomials. This allows to smooth the interpolation course. For third degree spline in each of the N-l intervals between adjoining nodes there exist:

Sj(x) = CLiX³ + biX² + c x + di where: i = 1... N-l

Having recreated the displacement and rotation curves, it is possible to use the thin-walled rod theory to determine the stresses and deformations in the cross-sections along the length of the rod. Below is the theory applied during the development of the algorithm to establish the numerical model taken from a published source [Piechnik]

The core of the thin-walled rod theory consists of the following hypothesis formulated by Vlasov:

• the middle surface deforms in the manner as if in each cross-sectional plane there was a rigid (but ideally tenuous for deformation in the direction perpendicular to the cross section) disc spanning along the middle line.

• Angular deformation at the points on the central surface are low

• Normal stresses σ_η of the surfaces parallel to the middle surface are negligible relative to the remaining two normal stresses σ_χ and σ₅

Fig. 30 Displacements in the global system

The following notation is used in connection with the global x, y, z system (main central axes):

• Parametric equation of the middle line y = yO) z = z(s)

Where s is the natural parameter

• Coordinates of the local system versors e [l,0,0] es[0, y(s), z(s)] en[0, -z(s), y(s)]

• Displacement coordinates of the R point u_R [u{x), v{x), w{x)]

Note that these coordinates depend only on the variable x, which is due to the fact that the deflection centre in each cross-section can be subject to different displacement. The independence of the remaining variables y and z is due to the fact that the displacement is connected with the translation of a rigid line.

• coordinates of the rotation vector

€[α(χ), φ(χ), ΰ(χ ]

In this case also all coordinates depend only on x. The rotation vector can be different in different cross-sections. The independence of the remaining variables is due to the fact that the rotation vector is constant in a given cross-section. The first coordinate is the angle of rotation (torsion) around any axis parallel to x, the second - the angle of rotation around any axis parallel to y, and the third respectively around any axis parallel to z.

• Coordinates of the deflection centre

R(x, y_Rl z_R)

• The coordinates of the main point of the zero fragmentary coordinate

Q [x, y_Q = yO = o), z_Q = z{s = 0)]

• Coordinates of the vector of displacement of any point

MM' [U(x, y, z), v(x, y, z), w(x, y, z)]

The variables can be used to express the following: u{x, y, z) = u(x) - w'(x)(z - ζ_β) - v x)(y - y_R) v(x ;, y, z = v(x) - a(x)(z - z_R) w(x, y, z) = w(x) - a(x)(y - y_R) The first equation above can be used to calculate the displacement of the cross-sectional centre of gravity and the displacement of the main point of the zero fragmentary coordinate in x direction.

The above equation in connection with ones not presented can be used to express the displacement of the centre of gravity by the displacement of the point Q: u₀ = v x, y_Q, z_Q) + w'(x)z_Q + v'(x)y_Q

Displacements in a local system

Displacement of any point M of the rigid disk is represented using coordinates of a local system. For this purpose, the following notation was used:

• Coordinates of the vector of displacement of any point

MM' [u(x, s), v(x, s), w(x, s)]

• Coordinates of the R point displacement vector

^UR [^URX .^x> ^s)> ^uRS(^x> ^s)> ^uRni.^x> ⁵)]

• Coordinates of the rotation vector and radius vector e[a(x), (x, s), d(x, s)] p[0, p_s(s), p_n(s)]

Consequently, the coordinates of the vector of displacement of the middle line in a local coordinate system can be expressed as: u_x(x, s) = u(x) - X(x, s)p_n(s) - v(x, s)p_s(s) u_s(x, s) = v(x) (s) + ( )z(s)— a(x)p_n(s) u_n(x, s) =—v(x)z(s) + w(x)y(s) + a(x)p_s(s) Summary of displacement

As a result of the above considerations, all of the projected displacement functions can be expressed as follows: i½(x, s) = u₀ (x)— '^wy(s)— w(x)z s) + a'(x)a)(s) u_s(x, s) = v(x)y(s) + w(x)i(s) - a(x)p_n(s) u_n(x, s) = -v(x)z(s) + w(x)y(s) + a(x)p_s s)

The first three components of the first of the above relations represent the displacement of rigid cross-section in the direction of the x axis. It can be stated that they implement the Bernoulli's principle for flat cross-sections. Consequently, the warping of the cross-sectional line is indicated by the last component: a'( )w(s).

The displacement functions are determined with an accuracy of four known functions of one variable x, namely u₀(x), v(x), w(x), a(x). These functions are the result of reading the respective values from the sensors of the device. The functions determine the following: u₀(x) - displacement of the centre of gravity along the x axis with the movement of the cross-section as a rigid disk v(x), w(x) - displacement of the deflection line points along the y and z axes a (x) - torsion angle of the cross-section around the x axis

Field of deformations at the points of the middle surface

Having established the field of displacements in the local system, the field of deformations can be determined for the points of the middle surface defined as: ε_χ(χ, s), e_s(x, S), Y_xs(X, S) The determination of the values of angular deformation of the surface was ignored in these considerations as too extensive to present. This value is determined using Navier's equation expressed as follows: da_x(x, s, n) ^ dz_xs(x, s, n) ^ dz_xn(x, s, n)

dx ds dn

As far as linear deformations are concerned, the first of these was calculated using Cauchy's equations and the functions of the assumed displacement field: du (x

ε_χ = * ' = -v"{x)y{s) - w"(x)zO) + α"(χ)ω(χ) + u'₀(x)

The second function as a consequence of the first Vlasov's hypothesis is identically equal to zero: du_s{x, s)

e_s(_.x, s) = — = 0

Field of stresses at the points of the middle surface

Normal stresses

Using the third Vlasov's hypothesis and the following Hooke's equations, the following equations can be obtained:

1

^£S = E i^as - ^{ν σ}χ + ^ση)] = 0

The above formulas ignore σ_η as negligible relative to the remaining normal stresses. By substituting o_s in the second equation with o_x the following is obtained: a_s = νσ_χ and by substituting it to the first equation and then calculating σ_χ the following is obtained: E

1— v

Expressing the linear deformations on the right-hand side of the equation by displacement function the following is obtained:

E

o_x(x, s) = - J [-v"(*)y(s) - w"( )z(s) + α"(χ)ω(_?) + '₀(*)]

■ \— vⁱ

Shear stresses

From the third Hooke's linear law the following can be obtained:

Field of displacements as a function of sectional forces

The relations not used in the description were used to directly obtain the differential equations for determining the displacement of the beam axis along x and the deflection and angles of deflection of bending axis in the xy and xz planes.

⁰ EA M_z(x)

v"(x)

EI₂

My(x)

w"{x) =

Ely

The torsion angle of each cross-section around the x axis is determined as follows:

M_Rx(x = -ΕΙ_ωα"'{χ) + GI_xa'(x)

The above equation is crucial as it allows to use the function of the torsion angle of each cross-section to determine the following:

• Warping of the cross-section, as described by the expression α'(χ)ω(5) • Bimoment Β_ω(χ) functions - belonging to the group of cross-sectional forces

• Deflection-torsion moment Μ_ω(χ) functions - belonging to the group of cross-sectional forces

The above relations allow to determine the values of the sectional forces acting on the respective thin-walled cross-section. This allows to obtain a hypothetical load condition necessary to determine the strengthening to analyse the reinforcement of the cross-section in an iterative manner. The calculations that analyse the mentioned calculation phase are performed with the use of a specially developed algorithm based on the finite element method.

Numerical calculations

Numerical calculations that are performed in the device according to the invention are based on the finite element method (FEM). This method is used to establish a discrete model of the structure, dividing it into parts referred to as finite elements. These parts connect to each other at points referred to as nodes. If the types and dimensions of the elements are specified, together with their mechanical properties and the manner of connection at the nodes, it is possible to obtain, on the basis of the relevant theoretical and numerical calculations, a specific solution of the strength-related problem. Depending on the type of structure (element), FEM distinguishes the respective defined elements. For example, in rod structures rods will be finite elements considered as one- dimensional. In the case of plates and shells, it will be a two-dimensional objects of different shapes, selected adequately to the problem. Three-dimensional elements are used to solve the problems involving solid bodies. Due to the possibility of the displacement of the structure, nodes are assigned a certain number of degrees of freedom. There is a distinction between fixed and movable degrees of freedom, as well as degrees of freedom with limited mounting. The degree of freedom is movable, if it is assigned to the component of node displacement to which no constraints (supports) are applied. If bilateral geometric constraints are applied to the component of node displacement, the degree of freedom of this component is fixed or stationary. Similarly, the limited degree of freedom allows slight displacement in the direction of the constraints.

In the case of the device according to the invention and to determine the actual load conditions and deformation, plate-disk elements with six degrees of freedom were used, supported by geometric nonlinearity. This gives the complete picture of the behaviour of the element under load, including critical load causing loss of stability of the element, as well as the loss of local stability. As the sensor of the position of the cross-section and the entire element do not consider buckling or local loss of stability of the section walls, the nonlinearity used in the FEM will be relatively insignificant. By contrast, in attempts to theoretically analyse the tested cross-section (i.e. to theoretically analyse application of additional load or increase of existing load), the nonlinearity used in the calculations will play a key role for the device. Therefore, the calculation mode can be represented by the chart shown in Fig. 31.

Fig. 32 shows the mode of measurements and calculations for the physical element subjected to analysis. Linear model was adopted as the first possible computing approach.

Another chart, shown in Fig. 33, presents the mode of theoretical analysis and possible simulation. It is characterised by a purely theoretical approach to the described element, not directly based on the carried out measurements. This allows to test any type of structural elements and to analyse the loss of stability for the element.

The method of geometric nonlinearity is based on the so-called co- rotational approach. This technique, in order to describe non-linear continuum, uses co-rotational algebra to separate the rigid body movement from the deformation of the element. Each material point P is defined by its position X. The current position after the deformation can be expressed as follows:

= X + u[X]

The deformation gradient is defined as: F[X] = Vx = / + Vu

The above equation (deformation gradient F[X]) can be expressed as the product of the rotation tensor R and tensor span of symmetrically and positively defined U:

F = RU

The co-rotational approach is very convenient approach since of the finite element matrix is not modified directly. Instead, only the part of the matrix derived from the nonlinearity is added. The Runge-Kutta method was used to solve the system of equations in the nonlinear approach. This method combines both the iterative and incremental method. The load Q is divided into increments AQj. Within each iteration increment is applied. The device according to invention employs a variant of the method with variable stiffness matrix.

Below is a short description of the method.

The deformation condition corresponding to displacements q₀ is not the actual state and therefore the internal forces are not in equilibrium with load Q. In other words, part of the load is not balanced. This part is denoted as Q₀ and calculated using the following equation:

Q₀ = (K^L + K»^L)q₀ - Q

Unbalanced forces Q₀ cause increase of the displacements equal to:

Changing the condition of displacement to q_t = q₀ + q_x

This condition is also not actual, so once again part of the load is not balanced. This section causes further increase of the displacement and changes the configuration of the calculated element so that resembles the actual one more accurately. This cycle is repeated until the increase of the unbalanced forces Q₀ and the increase of the displacement &q_n is acceptably low. In order to enable the above tasks, the control unit is equipped with an intuitive and practical graphical interface, based on the modern trends and taking into account the requirements for CAD software. The used technique of drawing, viewing and analysing in detail the defined numerical elements is based on the OpenGL graphic technology. There are maps of the displacement stress of the elements available, which significantly facilitates the understanding of the tackled problem. Below is a section of a map for a system of stresses:

Iterative process of selecting the strengthening

The final result of the calculations and the purpose of the device according to the invention, apart from purely theoretical analysis, is the ability to strengthen the areas of the cross-section to achieve the desired effect. This effect can be divided into the following parts:

• Reduced deflection

• Reduced rotation of the cross-section around the X axis

• Reduced stress in the cross-section.

The person that makes the analysis to parameterise his or hers requirements defines, in the PC, the restrictions (deflection, rotation, stress) by entering the appropriate parameters to the software. The calculation system, which is an iterative system, runs the calculations that are based on continuous stepwise approximation to final results.

The method can be divided into several phases of the calculation process. The first phase is the preparation of input data required in the later phases. The second phase is the phase of the iterative process of "thickening" finite elements (discrete model of the tested structure) at the locations as required by the imposed limits and restrictions. This process is based on the similarity of the results of nonlinear dependence of the dimensions of the cross-section on the values of the sectional forces in the finite element. The next phase of the calculations involves searching common fields and joining them to form larger area in order to determine the value of the strengthened area. Subsequently, the algorithm selects the thickness, size and location of the strengthening on the element. In this phase, the algorithm proceeds in accordance with the rules of the construction industry. Fig. 34 shows the chart of the discussed calculation phases expanded to include additional phases omitted in the discussion.

Iterative process of "thickening".

The starting point of the method is the assumption that the rigidity of the finite element considered in the numerical analysis is local rigidity. This fact allows to consider the strengthened element by analysing the respective finite elements. The consequence of the above assumption is the following calculation method:

• A load diagram is defined, for which the stresses in the strengthened cross-section of the element will be calculated. It will be a diagram of quasi-static loads including the following: self-weight load and equivalent load simulating the load condition of the tested element. The calculations exceed the range linear dependence and thus the principle of superposition of loads no longer applies. The sets of effects on the structure are considered.

• The values of the stresses calculated in the first iterative step are calculated for the entire element and accepted. This calculation is carried out due to the extreme values of sectional forces in the finite element (numerical model).

• The conditions restricting the stress in individual finite elements are determined. That is, the limit values are set that depend on the set limits of deflection or displacement occurring in the strengthened element. This process is a critical part of the algorithm.

• Appropriate parameters for individual finite elements, i.e. their thickness, are selected so that the conditions of the stress limits are met.

• The calculations for the structure are repeated, but this time they involve the strengthened structure, i.e. the structure with thickened finite elements.

• The result of these calculations is a new distribution of the internal forces in the calculated element, incompatible with the distribution, for which the rigidity of individual elements was previously established.

• New rigidity parameters of individual finite elements are determined for the new distribution of internal forces and the subsequent calculations are made for the structure.

• The procedure is repeated until the difference between the internal forces adopted to determine the rigidity and the values of force obtained as a result of the calculations becomes negligible for the determination of the stresses and displacements.

• The procedure is an iterative process, in most cases it rapidly converges.

Despite this fact, some procedures are used that accelerate this convergence.

The calculations carried out on the basis of this method showed that the very first calculation of the structure gives a good indication as to the value of the thickness of individual finite elements of the strengthened structure. The iteration No. 2-3 can be considered satisfactory. The algorithm has been designed to track the results of iteration. It may be the case that the iteration steps do not converge or oscillate (the graph of the convergence of iterations, similar to sine wave). The procedure implemented in the device procedure allows to shorten the iterative process and to even out the oscillating sequence of iterations.

Creating zones of common parameterisation

Zones creation is based on grouping of computing elements into common areas of similar geometric parameters (thickness) taking into account the deviation from the highest value. The algorithm attempts to select and merge zones with each other to create a single uniform area of maximum possible size. The process is shown in Fig. 37

The perforations referred to in Fig. 37 are places in the area of grouped zones of strengthening that may be omitted in strengthening. In other words, such element of strengthening will have zones with openings that will help to reduce the resulting weight of the strengthened structure. For elements of this type, it is of considerable importance in some circumstance.

The maximum of geometric values of the elements (their thickness) will not be the values that define the thickness of uniform zones, but only the reference values with corresponding weighting. The thickness of walls of the strengthening zone will be an averaged thickness. The device implements the following approach for averaging: the greater the value the greater factor is taken into account when calculating the thickness. This approach is expressed in the following equation:

^2 ^ ^71 ^

T = Α_Λ * * k + A₇ * * k +— h A_n * * k

t t t

_ T

t-zone where:

T - Parameter that specifies the impact of various finite elements on the thickness of the strengthening zone

A_n - Surface area of the n-th finite element t_n - Thickness of the n-th finite element tmax - Maximum thickness found in the analysed area k - Reduction factor usually assumed to be 1 tzone - Resulting averaged thickness of the analysed zone

A_c - Total surface area of the analysed zone

After grouping and setting the averaged geometric average value of the respective zone, the entire structure is recalculated and the input parameters are adjusted, if necessary.

Location and type of the proposed strengthening.

The final result is a graphical presentation of the proposed zones. The user can freely choose the shape covering the area defined by the algorithm. This shape will be interpreted by the program as the shape of strengthening with the set averaged thickness. The device creates the boundaries of the zones while maintaining the basic geometric shapes chosen by the user. Example 1.

The purpose of this example is to prove the correctness of the assumptions made for the devices, namely the correctness of recreating a physical element as a numerical number in the device. The correctness of the calculation results will be checked by verifying the correctness of the following parameters:

• Correspondence between the measured deflections and the deflection of the numerical model to which equivalent load is applied as selected by the algorithm.

• Correspondence of the measured rotation angles and the rotation angles of the numerical model.

The strengthening option will not be used in this example.

The considered item is a thin-walled I-beam (Fig. 36) with the cross- sectional dimensions of b=200 mm; h=300 mm and the thickness of the flanges and the web of t=5mm. The length of the tested beam is 5 m. A concentrated even load of 5 kl\l was applied to the beam at the distance of 2.5 m from the end of the beam. Fixed elements are mounted on both ends of the beam thus creating a connection with reduced susceptibility.

Five sensors were placed on the tested beam at equal intervals (every 1 m). The optical set was mounted on one end of the beam. The combination of beams with the fixed element was assumed to be rigid fastening of numerical element.

The numerical model assumed 0.05 m mesh of finite elements

Statistics for the tested beam Table 1.1

Description Value Unit

Length of the element 5 [m]

I-beam cross-section dimensions 5x200x300 [mm]

Concentrated load force 5 [kN] Point of applying load 2.5 [m]

Number of sensors mounted at 5 [pieces] regular intervals

Material parameters E 205 [GPa]

Material parameters v 0.3 [-]

Material parameters Re 215 [MPa]

Statistics of the generated numerical element Table 1.2

Description Value Unit

Number of nodes 1511 [-]

Element mesh size 0.05 [m]

Boundary conditions - start Fastening [-]

Boundary conditions - end Fastening [-]

The results of measurement part in the areas of mounting the sensors are summarised in the tables. The readout operations were performed in the conditions of actual behaviour of the construction.

The parameter indicating the warping angle (rotation of the cross-section around the axis of the rod) is the "Rotation angle RX". In this example, virtually no warping of the cross-section was observed (it was minimal).

Readout from the first sensor Table 1.3

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.00 [mm]

Vertical displacement UZ 0.14 [mm]

Rotation angle RX 0.0000 [rad]

Rotation angle RY 0.0002 [rad]

Readout from the second sensor Table 1.4

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.00 [mm]

Vertical displacement UZ -0.345 [mm]

Rotation angle RX 0.0000 [rad] Rotation angle RY -0.0001 [rad]

Readout from the third sensor Table 1.5

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.00 [mm]

Vertical displacement UZ -0.345 [mm]

Rotation angle RX 0.0000 [rad]

Rotation angle RY -0.0001 [rad]

Readout from the fourth sensor Table 1.6

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.00 [mm]

Vertical displacement UZ -0.14 [mm]

Rotation angle RX 0.0000 [rad]

Rotation angle RY 0.0002 [rad]

Readout from the fifth sensor Table 1.7

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.00 [mm]

Vertical displacement UZ 0.00 [mm]

Rotation angle RX 0.0000 [rad]

Rotation angle RY 0.0000 [rad]

The sensor data have been numerically processed to develop the numerical model of the tested element. This involved setting the properly parameterised cross-section (setting its shape and size) and parameterising the material used in the thin-walled cross-section. In addition, the boundary conditions needed to be defined. Because of the assumption that the existing conditions can be modelled as rigid constraint, this option has been selected as a boundary condition model. Below are the tables presenting the obtained calculation values together with an indication of the difference between the measurements and the theoretical values. As mentioned at the beginning of the description of the example, the purpose of the values calculated in this example is only to demonstrate the correctness of the algorithms that convert the sensor data to a numerical model. This is accomplished by verifying whether there is a difference between the measurements and calculations.

Readout values at the position of the first sensor Table 1.8

Readout values at the position of the second sensor Tabl

Readout values at the position of the third sensor Table 1.10

Description Value Unit Difference

Longitudinal displacement UX 0.00 [mm] 0.0%

Horizontal displacement UY 0.00 [mm] 0.0%

Vertical displacement UZ -0.342 [mm] 0.8%

Rotation angle RX 0.0000 [rad] 0.0%

Rotation angle RY 0.0001 [rad] 0.0% Readout values at the position of the fourth sensor Table 1.11

Readout values at the position of the fifth sensor Table 1.12

Summary of example 1.

The example shows negligible differences between the calculated values and the values measured by sensors. It should be considered that the accuracy is significantly affected by the discretisation of the structure into finite elements in the numerical model. The user can influence this parameter. For practical purposes, the assumed mesh size is 0.05 [m].

Example 2.

The purpose of this example, similarly to example 1, is to prove the correctness of the assumptions made for the devices, namely the correctness of recreating a physical element as a numerical number in the device. This example introduces an additional force acting perpendicular to the length of the beam in order to force noticeable buckling. The correctness of the calculation results will be checked by verifying the correctness of the following parameters:

• Correspondence of the measured rotation angles and the rotation angles of the numerical model. In this case, it will involve verifying the correspondence between the measured angles of buckling and the angles calculated by the measuring device.

The strengthening option will not be used in this example.

The considered item is a thin-walled I-beam (Fig. 37) with the cross- sectional dimensions of b=200 mm, h=300 mm and the thickness of the flanges and the web of t=5mm. The length of the tested beam is 5m. A concentrated even vertical load of 5 kN and a horizontal load of 1 kN were applied to the beam at the distance of 2.5 m from the end of the beam. Fixed elements are mounted on both ends of the beam, thus creating a connection with reduced susceptibility.

The numerical model assumed 0.05 m mesh of finite elements

Statistics for the tested beam Table 2.1 Description Value Unit

Length of the element 5 [m]

I-beam cross-section dimensions 5x200x300 [mm]

Concentrated vertical load force 5 _. [kN]

Concentrated horizontal load 1 [kN] force

Point of applying load 2.5 [m]

Number of sensors mounted at 5 [pieces] regular intervals

Material parameters E 205 [GPa]

Material parameters v 0.3 [-]

Material parameters Re 215 [MPa]

Statistics of the generated numerical element Table 2.2

Description Value Unit

Number of nodes 1511 [-]

Element mesh size 0.05 [m]

Boundary conditions - start Fastening [-]

Boundary conditions - end Fastening [-]

The parameter indicating the warping angle (rotation of the cross-section around the axis of the rod) is the "Rotation angle RX". Warping of the cross- section was observed in this example.

Readout from the first sensor Table 2.3

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.17 [mm]

Vertical displacement UZ -0.14 [mm]

Rotation angle RX 0.0011 [rad]

Rotation angle RY -0.0002 [rad] Readout from the second sensor Table 2.4

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY -0.43 [mm]

Vertical displacement UZ 0.344 [mm]

Rotation angle RX 0.0029 [rad]

Rotation angle RY -0.0001 [rad]

Readout from the third sensor Table 2.5

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.43 [mm]

Vertical displacement UZ -0.344 [mm]

Rotation angle RX 0.0029 [rad]

Rotation angle RY 0.0001 [rad]

Readout from the fourth sensor Table 2.6

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.17 [mm]

Vertical displacement UZ -0.14 [mm]

Rotation angle RX 0.0011 [rad]

Rotation angle RY 0.0002 [rad]

Readout from the fifth sensor Table 2.7

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.00 [mm]

Vertical displacement UZ 0.00 [mm]

Rotation angle RX 0.0000 [rad]

Rotation angle RY 0.0000 [rad] The sensor data have been numerically processed to develop the numerical model of the tested element. This involved setting the properly parameterised cross-section (setting its shape and size) and parameterising the material used in the thin-walled cross-section. In addition, the boundary conditions needed to be defined. Because of the assumption that the existing conditions can be modelled as rigid constraint, this option has been selected as a boundary condition model.

Below are the tables presenting the obtained calculation values together with an indication of the difference between the measurements and the theoretical values. As mentioned at the beginning of the description of the example, the purpose of the values calculated in this example is only to demonstrate the correctness of the algorithms that convert the sensor data to a numerical model. This is accomplished by verifying whether there is a difference between the measurements and calculations.

Readout values at the position of the first sensor Table 2.8

Readout values at the position of the second sensor Table 2.9

Description Value Unit Difference

Longitudinal displacement UX 0.00 [mm] 0.0%

Horizontal displacement UY 0.432 [mm] 0.5%

Vertical displacement UZ -0.343 [mm] 1.0%

Rotation angle RX 0.00287 [rad] 1.0%

Rotation angle RY -0.0001 [rad] 0.3% Readout values at the position of the third sensor Table 2.10

Readout values at the position of the fourth sensor Table 2.11

Readout values at the position of the fifth sensor Table 2.12

Summary of example 2.

The example shows practically negligible differences between the calculated values and the values measured by sensors. In engineering practice it is generally accepted that the difference up to 2% is acceptable for practical reasons, as it results from the assumptions in the numerical theory used to describe physical phenomena. The greatest difference between the readouts and calculations of 1.1% was observed in this example. This is a satisfactory result. It should be considered that the accuracy is significantly affected by the discretisation of the structure into finite elements in the numerical model. The user can influence this parameter. For practical purposes the assumed mesh size is 0.05 [m].

Example 3.

The purpose of this example is to prove the correctness of the assumptions made for the devices, namely the correctness of recreating a physical element as a numerical number in the device. Additionally, a system was introduced consisting of two elements, i.e. a beam and a column connected in the node with a butt connection. This arrangement is used to verify the applicability of the connection module in the numerical model. The example with the model of the butt connection will be used. The correctness of the calculation results will be checked by verifying the correctness of the following parameters:

The strengthening option will not be used in this example.

The considered item thin-walled I-beam and column (Fig. 38) with the cross-sectional dimensions of b=200 mm, h=300 mm and the thickness of the flanges and the web of t=5mm. The length of the tested beam is 6 m. Column height is 4 m. A concentrated even vertical load of 6kN was applied to the beam at the distance of 2 m from the end of the beam fastening. Fixed elements is mounted on one end of the beam, thus creating a connection with reduced susceptibility. The other end is a butt connection joining the beam and column to form a three-dimensional system. The column parameters are additionally parameterised in the device software. Five sensors were placed on the tested beam at equal intervals (every 1.2 m). The optical set was mounted on the fastened end of the beam. The combination of beams with the fixed element was assumed to be rigid fastening of numerical element.

The numerical model assumed 0.05 m mesh of finite elements. The mesh of equal size was adopted for both elements that is the beam and the butt connection.

Statistics for the tested beam Table 3.1

The results of measurement part in the areas of mounting the sensors are summarised in the tables. The readout operations were performed in the conditions of actual behaviour of the construction. The parameter indicating the warping of the cross-section (the rotation angle of the cross-section around the axis of the rod) is the "Rotation angle RX". In this example no warping of the cross-section was observed (it was negligible).

Sensors were placed starting from the fastening of the beam in the fixed element. The last sensor was placed at the beam-column contact point.

Readout from the first sensor Table 3.3

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.00 [mm]

Vertical displacement UZ -0.40 [mm]

Rotation angle RX 0.000 [rad]

Rotation angle RY -0.00045 [rad]

Readout from the second sensor Table 3.4

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.00 [mm]

Vertical displacement UZ -0.82 [mm]

Rotation angle RX 0.000 [rad]

Rotation angle RY -0.0002 [rad]

Readout from the third sensor Table 3.5

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.00 [mm]

Vertical displacement UZ -0.80 [mm]

Rotation angle RX 0.000 [rad]

Rotation angle RY 0.0001 [rad]

Readout from the fourth sensor Table 3.6

Description Value Unit

Longitudinal displacement UX 0.00 [mm] Horizontal displacement UY 0.00 [mm]

Vertical displacement UZ -0.47 [mm]

Rotation angle RX 0.000 [rad]

Rotation angle RY 0.0003 [rad]

Readout from the fifth sensor located at the column-beam contact point Table 3.

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.00 [mm]

Vertical displacement UZ -0.03 [mm]

Rotation angle RX 0.0000 [rad]

Rotation angle RY 0.0004 [rad]

The sensor data have been numerically processed to develop the numerical model of the tested element. This involved setting the properly parameterised cross-section (setting its shape and size) and parameterising the material used in the thin-walled cross-section. In addition, the boundary conditions needed to be defined. Because of the assumption that one of the existing conditions could be modelled as rigid constraint, this option has been selected as a boundary condition model for the beginning of the beam. For the end of the beam (beam-column contact point), the butt connection was selected from a library of available connections that was parameterised based on to the connection present in the structure.

Below are the tables presenting the obtained calculation values together with an indication of the difference between the measurements and the theoretical values.

Readout values at the position of the first sensor Table 3.8

Description Value Unit Difference

Longitudinal displacement UX 0.00 [mm] 0.0%

Horizontal displacement UY 0.00 [mm] 0.0%

Vertical displacement UZ -0.408 [mm] 0.5%

Rotation angle RX 0.000 [rad] 0.0%

Rotation angle RY -0.00045 [rad] 0.1% Readout values at the position of the second sensor Table 3.9

Readout values at the position of the third sensor Table 3.10

Readout values at the position of the fourth sensor Table 3.11

Readout values at the position of the fifth sensor Table 3.12

Description Value Unit Difference

Longitudinal displacement UX 0.00 [mm] 0.0%

Horizontal displacement UY 0.00 [mm] 0.0%

Vertical displacement UZ -0.0303 [mm] 1.0%

Rotation angle RX 0.0000 [rad] 0.0%

Rotation angle RY 0.000402 [rad] 0.5% Summary of example 3.

The example shows practically negligible differences between the calculated values and the values measured by sensors. In engineering practice, it is generally accepted that the difference up to 2% is acceptable for practical reasons, as it results from the assumptions in the numerical theory used to describe physical phenomena. The greatest difference between the readouts and calculations of 1.0% was observed in this example. This is a satisfactory result.

Example 4.

The purpose of this example is to prove the correctness of the assumptions made for the devices, namely the correctness of recreating a physical element as a numerical number in the device. The analysis includes a system consisting of two elements, i.e. a beam and a column connected in the node with a butt connection. This arrangement is used to verify the applicability of the connection module in the numerical model. The example with the model of the butt connection will be used. This example introduces an additional force acting perpendicular to the length of the beam in order to force noticeable buckling. The correctness of the calculation results will be checked by verifying the correctness of the following parameters:

The strengthening option will not be used in this example.

The considered item thin-walled I-beam and column (Fig. 39) with the cross-sectional dimensions of b=200 mm, h=300 mm and the thickness of the flanges and the web of t=5 mm. The length of the tested beam is 6 m. Column height is 4 m. An even concentrated vertical load of 6kN and a horizontal load of 1 kN were applied to the beam at the distance of 2 m from the end of the beam fastening. Fixed element is mounted on one end of the beam, thus creating a connection with reduced susceptibility. The other end is a butt connection joining the beam and column to form a three-dimensional system. The column parameters are additionally parameterised in the device software.

Five sensors were placed on the tested beam at equal intervals (every 1.2 m). The optical set was mounted on the fastened end of the beam. The combination of beams with the fixed element was assumed to be rigid fastening of numerical element.

The numerical model assumed 0.05 m mesh of finite elements. The mesh of equal size was adopted for both elements, i.e. the beam and the butt connection.

Statistics for the tested beam Table 4.1

Description Value Unit

Beam length 6 [m]

Column height 4 [m]

Column and beam connection Butt connection [-]

I-beam cross-section dimensions 5x200x300 [mm]

Concentrated vertical load force 6 [kN]

Concentrated horizontal load 1 [kN] force

Point of applying load 2.0 [m]

Number of sensors mounted at 5 [pieces] regular intervals

Material parameters E 205 [GPa]

Material parameters v 0.3 [-]

Material parameters Re 215 [MPa] Statistics of the generated numerical element Table 4.2

The parameter indicating the warping of the cross-section (the rotation angle of the cross-section around the axis of the rod) is the "Rotation angle RX". In this example no warping of the cross-section was observed (it was negligible).

Readout from the first sensor Table 4.3

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.55 [mm]

Vertical displacement UZ -0.38 [mm]

Rotation angle RX 0.0027 [rad]

Rotation angle RY -0.00045 [rad]

Readout from the second sensor Table 4.4

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 1.43 [mm]

Vertical displacement UZ -0.82 [mm]

Rotation angle RX 0.0062 [rad]

Rotation angle RY -0.0002 [rad] Readout from the third sensor Table 4.5

Readout from the fourth sensor Table 4.6

Readout from the fifth sensor located at the column-beam contact point Table 4.7

Readout values at the position of the first sensor Table 4.8

Description Value Unit Difference

Longitudinal displacement UX 0.00 [mm] 0.0%

Horizontal displacement UY 0.552 [mm] 0.4%

Vertical displacement UZ -0.385 [mm] 1.3%

Rotation angle RX 0.0027 [rad] 0.2%

Rotation angle RY -0.00045 [rad] 0.3%

Readout values at the position of the second sensor Table 4.9

Description Value Unit Difference

Longitudinal displacement UX 0.00 [mm] 0.0%

Horizontal displacement UY 1.427 [mm] 0.2%

Vertical displacement UZ -0.819 [mm] 0.2%

Rotation angle RX 0.0061 [rad] 1.6%

Rotation angle RY -0.0002 [rad] 0.2%

Readout values at the position of the third sensor Table 4.10

Description Value Unit Difference

Longitudinal displacement UX 0.00 [mm] 0.0%

Horizontal displacement UY 1.97 [mm] 0.1%

Vertical displacement UZ -0.802 [mm] 0.3%

Rotation angle RX 0.0065 [rad] 1.5%

Rotation angle RY 0.0002 [rad] 0.1%

Readout values at the position of the fourth sensor Table 4.11

Description Value Unit Difference

Longitudinal displacement UX 0.00 [mm] 0.0%

Horizontal displacement UY 2.26 [mm] 0.5% Vertical displacement UZ -0.477 [mm] 0.6%

Rotation angle RX 0.00508 [rad] 1.4%

Rotation angle RY 0.0003 [rad] 0.1%

Readout values at the position of the fifth sensor Table 4.12

Description Value Unit Difference

Longitudinal displacement UX 0.00 [mm] 0.0%

Horizontal displacement UY 2.104 [mm] 0.2%

Vertical displacement UZ -0.0208 [mm] 0.7%

Rotation angle RX 0.00089 [rad] 1.1%

Rotation angle RY 0.0004 [rad] 0.1%

Summary of example 4.

The example shows practically negligible differences between the calculated values and the values measured by sensors. The greatest difference between the readouts and calculations of 1.6% was observed in this example. This is a satisfactory result.

Example 5.

The purpose of this example is to demonstrate the correctness of the designated strengthening of the tested element subjected to concentrated load. The correctness of the calculation results will be checked by verifying the correctness of the following parameters:

• Measured deflection of the strengthened element (thickened parts of the structural element) and correspondence of the measured deflections with the set limits.

• Measurement of the angles of rotation of the strengthened element and the correspondence of the measured angle with the set limit.

The considered item is a thin-walled I-beam (Fig. 40) with the cross- sectional dimensions of b=200 mm, h=300 mm and the thickness of the flanges and the web of t=5mm. The length of the tested beam is 5 m. A concentrated even vertical load of 5 kN and a horizontal load of 2kN were applied to the beam at the distance of 2.5 m from the end of the beam. Fixed elements are mounted on both ends of the beam, thus creating a connection with reduced susceptibility.

In order to force strengthening, limits for the values given in Table 5.8- 5.11 were set. This should result in thickening of the cross-sections in the zones of greatest strain.

The numerical model assumed 0.05 m mesh of finite elements

Statistics for the tested beam Table 5.1

Description Value Unit

Length of the element 5 [m]

I-beam cross-section dimensions 5x200x300 [mm]

Concentrated vertical load force 5 [kN]

Concentrated horizontal load 2 [kN] force

Point of applying load 2.5 [m]

Number of sensors mounted at 5 [pieces] regular intervals

Material parameters E 205 [GPa]

Material parameters v 0.3 [-]

Material parameters Re 215 [MPa]

Statistics of the generated numerical element Table 5.2

Description Value Unit

Number of nodes 1511 [-]

Element mesh size 0.05 [m]

Boundary conditions - start Fastening [-]

Boundary conditions - end Fastening [-] The results of measurement part in the areas of mounting the sensors are summarised in the tables. The readout operations were performed in the conditions of actual behaviour of the construction.

Readout from the fourth sensor Table 5.6 Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.34 [mm]

Vertical displacement UZ -0.14 [mm]

Rotation angle RX 0.0021 [rad]

Rotation angle RY 0.0002 [rad]

Readout from the fifth sensor Table 5.7

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 0.00 [mm]

Vertical displacement UZ 0.00 [mm]

Rotation angle RX 0.0000 [rad]

Rotation angle RY 0.0000 [rad]

The tables below present the limits for the respective section (in this case, the location of the sensors)

Limit for the first sensor Table 5.8

Description Value Unit

Horizontal displacement UY 0.17 [mm]

Rotation angle RX 0.0011 [rad]

Limit for the second sensor Table 5.9

Description Value Unit

Horizontal displacement UY 0.43 [mm]

Rotation angle RX 0.0029 [rad]

Limit for the third sensor Table 5.10

Description Value Unit

Horizontal displacement UY 0.43 [mm]

Rotation angle RX 0.0029 [rad] Limit for the fourth sensor Table 5.11

After the calculations, the device suggested to increase the thickness of the bottom and top strips to the value of 10 mm over the entire length of the rod. The suggested strengthening was introduced and subjected to a load with the initial values, i.e. a vertical force = 5 klM and a horizontal force of 2 kN.

Below are the tables presenting the obtained values together with the indication whether readout for the respective cross-section has exceeded the set limits.

Readout values at the position of the first sensor Table 5.12

Description Value Unit Verified

Horizontal displacement UY 0.17 [mm] YES

Rotation angle RX 0.001 [rad] YES

Readout values at the position of the second sensor Table 5.13

Description Value Unit Verified

Horizontal displacement UY 0.43 [mm] YES

Rotation angle RX 0.0027 [rad] YES

Readout values at the position of the third sensor Table 5.14

Description Value Unit Verified

Horizontal displacement UY 0.43 [mm] YES

Rotation angle RX 0.0027 [rad] YES Readout values at the position of the fourth sensor Table 5.15

Summary of example 5.

As a result of the conducted tests, it was found that the strengthening suggested by the device has satisfied its purpose. The set limits were not exceeded for any of the limited values in the cross-sections. In addition, it was found that the manner of strengthening is in line with engineering common sense and is possible to implement.

Example 6.

The purpose of this example is to demonstrate the correctness of the designated strengthening of the tested element subjected to concentrated load. The analysis includes a system consisting of two elements, i.e. a beam and a column connected in the node with a butt connection. This example introduces an additional force acting perpendicular to the length of the beam in order to force noticeable buckling. The correctness of the calculation results will be checked by verifying the correctness of the following parameters:

The considered item thin-walled I-beam and column (Fig. 41) with the cross-sectional dimensions of b=200 mm, h=300 mm and the thickness of the flanges and the web of t=5mm. The length of the tested beam is 6 m. Column height is 4 m. An even concentrated vertical load of 6 kN and a horizontal load of 2 kN were applied to the beam at the distance of 2 m from the end of the beam fastening. Fixed element is mounted on one end of the beam thus creating a connection with reduced susceptibility. The other end is a butt connection, joining the beam and column to form a three-dimensional system. The column parameters are additionally parameterised in the device software.

In order to force strengthening, limits for the values given in Table 6.8- 6.12 were set. This should result in thickening of the cross-sections in the zones of greatest strain.

Statistics for the tested beam Table 6.1

Description Value Unit

Beam length 6 [m] -

Column height 4 [m]

Column and beam connection Butt connection [-]

I-beam cross-section dimensions 5x200x300 [mm]

Concentrated vertical load force 6 [kN]

Concentrated horizontal load 2 [kN] force

Point of applying load 2.0 [m]

Number of sensors mounted at 5 [pieces] regular intervals

Material parameters E 205 [GPa]

Material parameters v 0.3 [-]

Material parameters Re 215 [MPa] Statistics of the generated numerical element Table 6.2

Readout from the first sensor Table 6.3

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 1.10 [mm]

Vertical displacement UZ -0.38 [mm]

Rotation angle RX 0.0053 [rad]

Rotation angle RY -0.0004 [rad]

Readout from the second sensor Table 6.4

Description Value Unit

Longitudinal displacement UX 0,00 [mm]

Horizontal displacement UY 2.86 [mm]

Vertical displacement UZ -0.811 [mm]

Rotation angle RX 0.012 [rad]

Rotation angle RY -0.0002 [rad] Readout from the third sensor Table 6.5

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 3.95 [mm]

Vertical displacement UZ -0.80 [mm]

Rotation angle RX 0.013 [rad]

Rotation angle RY 0.0002 [rad]

Readout from the fourth sensor Table 6.6

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 4.51 [mm]

Vertical displacement UZ -0.48 [mm]

Rotation angle RX 0.010 [rad]

Rotation angle RY 0.0003 [rad]

Readout from the fifth sensor Table 6.7

Description Value Unit

Longitudinal displacement UX 0.00 [mm]

Horizontal displacement UY 4.21 [mm]

Vertical displacement UZ -0.03 [mm]

Rotation angle RX 0.0017 [rad]

Rotation angle RY 0.0004 [rad]

Limit for the first sensor Table 6.8

Description Value Unit

Horizontal displacement UY 0.55 [mm]

Horizontal displacement UZ -0.38 [mm]

Rotation angle RX 0.0027 [rad] Limit for the second sensor Table 6.9

Description Value Unit

Horizontal displacement UY 1.43 [mm]

Horizontal displacement UZ -0.82 [mm]

Rotation angle RX 0.0062 [rad]

Limit for the third sensor Table 6.10

Description Value Unit

Horizontal displacement UY 1.97 [mm]

Horizontal displacement UZ -0.80 [mm]

Rotation angle RX 0.0066 [rad]

Limit for the fourth sensor Table 6.11

Description Value Unit

Horizontal displacement UY 2.41 [mm]

Horizontal displacement UZ -0.48 [mm]

Rotation angle RX 0.0051 [rad]

Limit for the fifth sensor Table 6.12

Description Value Unit

Horizontal displacement UY 2.71 [mm]

Horizontal displacement UZ -0.03 [mm]

Rotation angle RX 0.0011 [rad]

After the calculations, the device suggested to increase the thickness of the bottom and top strips to the value of 15 mm over the length of 0.5 m, starting from the fastened support. The suggested strengthening was introduced and subjected to a load with the initial values, i.e. a vertical force of 6 kN and a horizontal force of 2 kN. Below are the tables presenting the obtained values, together with the indication whether readout for the respective cross-section has exceeded the set limits.

Readout values at the position of the first sensor Table 6.13

Readout values at the position of the second sensor Table 6.14

Readout values at the position of the third sensor Table 6.15

Readout values at the position of the fourth sensor Table 6.16

Readout values at the position of the fifth sensor Table 6.17

Description Value Unit Verified

Horizontal displacement UY 2.71 [mm] YES

Horizontal displacement UZ -0.029 [mm] YES

Rotation angle RX 0.0010 [rad] YES Summary of example 6.

Claims

Patent claims

1. A measurement and calculation device for testing displacement of open and closed thin-walled steel elements in order to find a solution to strengthen the element that has undergone significant deformation, characterised in that it consists of interconnected PC (1) with software, measurement card (2) and optical set (3) with laser beam connected to the measurement card (2), and at least three rotation sensors (4) with accelerometer, which are placed on the tested element (5; 10), whereas:

3) the rotation sensor (4) consists of an accelerometer (6) located on a mounting platform consisting of a holding table (7) and a telescopic part (8) that slides out of it, whereas the holding table (7) base that is in contact with the tested element (5) and the telescopic part (8) is equipped with fastening magnets (9);

4) the optical set (3) consists of:

c) base part located on a stabiliser on the tested element (5; 10) and consisting of a photoelectric sensor (25; 26) and high-definition camera (27) with red guiding spot light;

d) movable part consisting of a measurement board (30) mounted on the stabiliser in the measuring points of deflection of the tested element (5; 10).

2. Measurement and calculation device according to claim 1 characterised in that the telescopic part (8) consists of the first arm (12) and the second arm (13) that slides out of the first arm, on which the fastening magnet (9) is mounted.