Laboratory and Numerical Analysis of Steel Cold-Formed Sigma Beams Retrofitted by Bonded CFRP Tapes—Extended Research

The presented research is a part of a broader study of strengthening methods closely associated with cold-formed sigma steel beams with tapes made of Carbon Fiber Reinforcement Polymer/Plastic (CFRP). The presented results are a continuation and extension of the tests described in previous work by the authors and refer to high-slenderness thin-walled steel sigma beams subjected to a significant large rotation. The main idea of this expanded study was to identify the effectiveness of CFRP tapes with respect to different locations, namely at a bottom-tensioned or upper-compressed flange. Six beams with a cross-section of an Σ140 × 70 × 2.5 profile by “Blachy Pruszyński” and made of S350GD steel with a span of L = 270 cm were tested in the four-point bending scheme. Two beams, taken as reference, were tested without reinforcement. The remaining beams were reinforced with the use of a 50-mm wide and 1.2-mm thick Sika CarboDur S512 CFRP tape, with two beams reinforced by placing the tape on the upper flange and two with tape located on the bottom flange. The CFRP tape was bonded directly to the beams (by SikaDur®-30 adhesive). Laboratory tests were aimed at determining the impact of the use of composite tapes on the limitation of displacements and deformations of thin-walled structures. In order to perform a precise measurement of displacement, which is, in the case of beams subjected to large rotations, a very difficult issue in itself, the Tritop system and two coupled lenses of the Aramis system were used. Electrofusion strain gauges were used to measure the deformation. In the next step, numerical models of the analyzed beams were developed in the Abaqus program. Good compliance of the results of laboratory tests and numerical analyses was achieved. The obtained results confirm the beneficial effect of the use of tapes (CFRP) on the reduction in displacements and deformations of steel cold-formed elements.


Introduction
Designers of steel structures constantly strive to design structures that are economical, safe and quick to build up. These requirements are satisfied by cold-formed, thin-walled steel structural elements, and therefore they are used more and more often [1]. With reference to Vlasov's definition, a bar can be contemplated as thin-walled if the wall thickness is approximately eight times smaller than the longest distance which is registered along the center line (between two extreme points located concern the element with a low cross-section (140 × 70 × 2.5 mm) characterized by the global bucking mode in the form of a large rotation. In the first stage of the studies [1], a good qualitative and quantitative correlation was obtained between the numerical and experimental results; the aim of these studies was also to obtain satisfactory results despite the change of geometric parameters of the structure cross-section. Moreover, the authors set themselves a goal to check whether the influence of CFRP tapes on the structural behavior for the low cross-section sigma beams (characterized by a global form of deformation) will be as significant as in the case of high cross-section sigma beams [1].

Laboratory Tests
Six thin-walled steel beams with a section of Σ140 × 70 × 2.5 [19], a length of 300 cm and made of S350GD steel, were included in the study. The geometrical characteristics of the cross-section are given in Figure 1 and in Table 1.  In the table above, FA is the cross-sectional area, eyy-the distance of the profile center of gravity with respect to the z axis, ezz-the distance of the profile center of with respect to the y axis, Jy-the moment of inertia of the section with respect to the y axis, Jz-the moment of inertia of the section with respect to the z axis, Wy-second modulus of area with respect to the y axis (minimum), Wzsecond modulus of area with respect to the z axis (minimum), iy-radius of gyration with respect to the y axis, and iz-radius of gyration with respect to the z axis.
Four of the tested beams were reinforced with CFRP tapes glued to different surfaces: two beams (MB1D, MB2D) were reinforced with CFRP tape located on the outer plane of the bottom flange, whereas two beams (MB1G, MB2G) had reinforcement placed on the upper flange (on the inner plane). Two beams (MB1R, MB2R-reference beams) were also tested without reinforcement. The arrangement of the tapes on the beams section is shown in Figure 2a).  In the table above, FA is the cross-sectional area, e yy -the distance of the profile center of gravity with respect to the z axis, e zz -the distance of the profile center of with respect to the y axis, J y -the moment of inertia of the section with respect to the y axis, J z -the moment of inertia of the section with respect to the z axis, W y -second modulus of area with respect to the y axis (minimum), W z -second modulus of area with respect to the z axis (minimum), i y -radius of gyration with respect to the y axis, and i z -radius of gyration with respect to the z axis.
Four of the tested beams were reinforced with CFRP tapes glued to different surfaces: two beams (MB1D, MB2D) were reinforced with CFRP tape located on the outer plane of the bottom flange, whereas two beams (MB1G, MB2G) had reinforcement placed on the upper flange (on the inner plane). Two beams (MB1R, MB2R-reference beams) were also tested without reinforcement. The arrangement of the tapes on the beams section is shown in Figure 2a). Laboratory tests were carried out to determine the strength parameters of steel. The research was carried out using five samples cut out from the tested thin-walled profiles. The size and shape of the samples resulted from the guidelines of PN-EN ISO 6892-1: 2009 [20]. The biaxial extensometer was used to measure the longitudinal and transverse deformation of the samples. On the basis of conducted tests, the steel yield point was determined (fy = 418.5 MPa), as well as Young's modulus (E = 201.8 GPa) and Poisson's ratio (ν = 0.28.) The tested beams were reinforced with Sika CarboDurS carbon fiber strips (CFRP), 50-mm wide and 1.2-mm thick. The CFRP tapes applied on the beams were composed of a unidirectional arrangement of carbon fibers sunk in an epoxy matrix and are characterized by an anisotropic structure. The conducted research allowed to determine the Poisson's ratio ν = 0.308 and the Young's modulus E = 142 GPa of the tape used. More details on the parameters of the composite tape are described in [21]. To connect the thin-walled steel beams with the CFRP tape, specialized SikaDur ® -30 adhesive was used. The most important features of this adhesive include the modulus of elasticity in compression equal to 9600 MPa, minimum compressive strength (75 MPa after 7 days), minimum tensile strength (26 MPa after 7 days), steel deboning (21 MPa after 7 days), shear strength (minimum 16 MPa) and 0.04% shrinkage.
The first stage of laboratory work was the preparation of samples for testing. Before laboratory tests, all beams were degreased and matted with medium grain sandpaper and cleaned in the areas where composite tapes were planned to be applied. The beams were then reinforced by gluing the CFRP tapes. In the study, 207.5-cm long tapes were used, with an effective anchorage length of 30 cm. The effective anchorage length was determined as described in [1]. The tapes were glued to the beams with 1.3-mm thick adhesive. The thickness of the adhesive layer was based on studies conducted by Kowal [22].
At the time of the laboratory tests, the strain was measured using three electrofusion strain gauges (TENMEX TFs-10, resistance 120 Ω ± 0.2%, Lodz, Poland). All electrofusion strain gauges (T1, T2, T3) were located in the middle of the span of each of the tested beams. Figure 2b shows the arrangement of strain gauges on the beam cross-section. Optical measuring systems were also used during the tests. They enable accurate measurement of the displacement of test samples undergoing large rotations. In this study, a special arrangement of two Aramis measuring lenses (GOM, GmbH, Braunschweig, Germany) and a Tritop machine (GOM, GmbH, Braunschweig, Germany) was used. Before the tests, white chalk was applied to the surface of the beams to make it extra matt so that it would not reflect light. Appropriate measuring points with a diameter of 5 mm were glued to the middle of the beam span. The point symbols are shown in Figure 2b. The Tritop system enables the assessment of initial geometric imperfections. The measurement points also enable the generation of one reference coordinate system for two coupled devices in the Aramis system [23]. They were also used for displacements analyses of the tested beams in the GOM Correlate program (GOM, GmbH, Braunschweig, Germany). The method and scope of the application of the Tritop and Aramis systems are discussed in detail in [23]. Laboratory tests were carried out to determine the strength parameters of steel. The research was carried out using five samples cut out from the tested thin-walled profiles. The size and shape of the samples resulted from the guidelines of PN-EN ISO 6892-1: 2009 [20]. The biaxial extensometer was used to measure the longitudinal and transverse deformation of the samples. On the basis of conducted tests, the steel yield point was determined (fy = 418.5 MPa), as well as Young's modulus (E = 201.8 GPa) and Poisson's ratio (ν = 0.28.) The tested beams were reinforced with Sika CarboDurS carbon fiber strips (CFRP), 50-mm wide and 1.2-mm thick. The CFRP tapes applied on the beams were composed of a unidirectional arrangement of carbon fibers sunk in an epoxy matrix and are characterized by an anisotropic structure. The conducted research allowed to determine the Poisson's ratio ν = 0.308 and the Young's modulus E = 142 GPa of the tape used. More details on the parameters of the composite tape are described in [21]. To connect the thin-walled steel beams with the CFRP tape, specialized SikaDur ® -30 adhesive was used. The most important features of this adhesive include the modulus of elasticity in compression equal to 9600 MPa, minimum compressive strength (75 MPa after 7 days), minimum tensile strength (26 MPa after 7 days), steel deboning (21 MPa after 7 days), shear strength (minimum 16 MPa) and 0.04% shrinkage.
The first stage of laboratory work was the preparation of samples for testing. Before laboratory tests, all beams were degreased and matted with medium grain sandpaper and cleaned in the areas where composite tapes were planned to be applied. The beams were then reinforced by gluing the CFRP tapes. In the study, 207.5-cm long tapes were used, with an effective anchorage length of 30 cm. The effective anchorage length was determined as described in [1]. The tapes were glued to the beams with 1.3-mm thick adhesive. The thickness of the adhesive layer was based on studies conducted by Kowal [22].
At the time of the laboratory tests, the strain was measured using three electrofusion strain gauges (TENMEX TFs-10, resistance 120 Ω ± 0.2%, Lodz, Poland). All electrofusion strain gauges (T1, T2, T3) were located in the middle of the span of each of the tested beams. Figure 2b shows the arrangement of strain gauges on the beam cross-section. Optical measuring systems were also used during the tests. They enable accurate measurement of the displacement of test samples undergoing large rotations. In this study, a special arrangement of two Aramis measuring lenses (GOM, GmbH, Braunschweig, Germany) and a Tritop machine (GOM, GmbH, Braunschweig, Germany) was used. Before the tests, white chalk was applied to the surface of the beams to make it extra matt so that it would not reflect light. Appropriate measuring points with a diameter of 5 mm were glued to the middle of the beam span. The point symbols are shown in Figure 2b. The Tritop system enables the assessment of initial geometric imperfections. The measurement points also enable the generation of one reference coordinate system for two coupled devices in the Aramis system [23]. They were also used for displacements analyses of the tested beams in the GOM Correlate program (GOM, GmbH, Braunschweig, Germany). The method and scope of the application of the Tritop and Aramis systems are discussed in detail in [23].
A four-point bending test was carried out using a test stand developed for this purpose. The assumed support spacing was 270 cm and load spacing was 135 cm (Figure 3). A four-point bending test was carried out using a test stand developed for this purpose. The assumed support spacing was 270 cm and load spacing was 135 cm (Figure 3). All tests were carried out in a certified laboratory at Lublin University of Technology. The process of beams loading was carried out with a Zwick and Roel testing machine (ZwickRoell GmbH and Co. KG, Ulm, Germany). Figure 4a shows a photo of the hinge screw connection at the support of the test stand. Such a connection allows free rotation in the beam plane and thus it is possible to obtain the so-called fork support. Additionally, to prevent local damage of the analyzed beams at the points of application of the concentrated force from the testing machine, a hot-rolled steel C100 profile (100-mm long) was used. This allowed the load to be distributed over the entire width of the upper flange ( Figure 4b). The process of loading the beams was controlled with an extendable piston press (speed 1 mm/min), registering the force every 0.01 s.  It should be pointed out that beams made of low cross-sections (Σ140 × 70 × 2.5), tested at this stage of study, are characterized by different structural behaviors than those observed in the case of high cross-section (Σ200 × 70 × 2) beams described in [1]. In the high beams [1], the deformation manifested by the opening of the beam's cross-section and the debonding failure mode. A significant lifting of the upper beam flange was observed between the load application points. At the same time, the cross-section moved out of the vertical plane of the beam. In addition, local deformations of the upper flange were observed in the places where concentrated forces were applied. There was no damage to the lower flange. In [1], in all high beams the tape was debonded at a load of about 25 kN. Whereas, in case of 140-mm cross-section high beams, described in this part of the study, a totally different failure mechanism appeared. All the beams were characterized by a global form of All tests were carried out in a certified laboratory at Lublin University of Technology. The process of beams loading was carried out with a Zwick and Roel testing machine (ZwickRoell GmbH and Co. KG, Ulm, Germany). Figure 4a shows a photo of the hinge screw connection at the support of the test stand. Such a connection allows free rotation in the beam plane and thus it is possible to obtain the so-called fork support. Additionally, to prevent local damage of the analyzed beams at the points of application of the concentrated force from the testing machine, a hot-rolled steel C100 profile (100-mm long) was used. This allowed the load to be distributed over the entire width of the upper flange ( Figure 4b). The process of loading the beams was controlled with an extendable piston press (speed 1 mm/min), registering the force every 0.01 s.

Materials 2020, 13, x FOR PEER REVIEW 5 of 16
A four-point bending test was carried out using a test stand developed for this purpose. The assumed support spacing was 270 cm and load spacing was 135 cm ( Figure 3). All tests were carried out in a certified laboratory at Lublin University of Technology. The process of beams loading was carried out with a Zwick and Roel testing machine (ZwickRoell GmbH and Co. KG, Ulm, Germany). Figure 4a shows a photo of the hinge screw connection at the support of the test stand. Such a connection allows free rotation in the beam plane and thus it is possible to obtain the so-called fork support. Additionally, to prevent local damage of the analyzed beams at the points of application of the concentrated force from the testing machine, a hot-rolled steel C100 profile (100-mm long) was used. This allowed the load to be distributed over the entire width of the upper flange ( Figure 4b). The process of loading the beams was controlled with an extendable piston press (speed 1 mm/min), registering the force every 0.01 s.  It should be pointed out that beams made of low cross-sections (Σ140 × 70 × 2.5), tested at this stage of study, are characterized by different structural behaviors than those observed in the case of high cross-section (Σ200 × 70 × 2) beams described in [1]. In the high beams [1], the deformation manifested by the opening of the beam's cross-section and the debonding failure mode. A significant lifting of the upper beam flange was observed between the load application points. At the same time, the cross-section moved out of the vertical plane of the beam. In addition, local deformations of the upper flange were observed in the places where concentrated forces were applied. There was no damage to the lower flange. In [1], in all high beams the tape was debonded at a load of about 25 kN. Whereas, in case of 140-mm cross-section high beams, described in this part of the study, a totally different failure mechanism appeared. All the beams were characterized by a global form of It should be pointed out that beams made of low cross-sections (Σ140 × 70 × 2.5), tested at this stage of study, are characterized by different structural behaviors than those observed in the case of high cross-section (Σ200 × 70 × 2) beams described in [1]. In the high beams [1], the deformation manifested by the opening of the beam's cross-section and the debonding failure mode. A significant lifting of the upper beam flange was observed between the load application points. At the same time, the cross-section moved out of the vertical plane of the beam. In addition, local deformations of the upper flange were observed in the places where concentrated forces were applied. There was no damage to the lower flange. In [1], in all high beams the tape was debonded at a load of about  25 kN. Whereas, in case of 140-mm cross-section high beams, described in this part of the study, a totally different failure mechanism appeared. All the beams were characterized by a global form of destruction. The beams suffered both deflection and a significant rotation, so they were subject to failure in the form of lateral-torsional buckling. No local failure was observed. Only local scratches appeared on the top flange, where the load was applied. In one of the reinforced beams, the CFRP tape was partially subjected to debonding. The way the beam works and its deformation appearing at a load of 25 kN are presented in Figure 5.
Materials 2020, 13, x FOR PEER REVIEW 6 of 16 destruction. The beams suffered both deflection and a significant rotation, so they were subject to failure in the form of lateral-torsional buckling. No local failure was observed. Only local scratches appeared on the top flange, where the load was applied. In one of the reinforced beams, the CFRP tape was partially subjected to debonding. The way the beam works and its deformation appearing at a load of 25 kN are presented in Figure 5. Examples of the displacements of measuring points of the selected sample in the middle of its span obtained with a load of 25 kN are presented in Figure 6.  Examples of the displacements of measuring points of the selected sample in the middle of its span obtained with a load of 25 kN are presented in Figure 6. destruction. The beams suffered both deflection and a significant rotation, so they were subject to failure in the form of lateral-torsional buckling. No local failure was observed. Only local scratches appeared on the top flange, where the load was applied. In one of the reinforced beams, the CFRP tape was partially subjected to debonding. The way the beam works and its deformation appearing at a load of 25 kN are presented in Figure 5. Examples of the displacements of measuring points of the selected sample in the middle of its span obtained with a load of 25 kN are presented in Figure 6.  During the tests, the tape was observed to be debonded at the steel-glue surface in only one beam. The debonding occurred only at one side of the composite tape (CFRP) on ca. 10% of its length and appeared with a load exceeding 30 kN.
An example diagram of the load-strain relationship obtained from readings from an electrofusion strain gauge (TENMEX TFs-10, resistance 120 Ω ± 0.2%, Lodz, Poland) for different CFRP tape reinforcement locations is shown in Figure 7, respectively. The diagrams of the load-strain relationship based on the readings of the remaining strain gauges (T1 and T2) are shown in the further figures. During the tests, the tape was observed to be debonded at the steel-glue surface in only one beam. The debonding occurred only at one side of the composite tape (CFRP) on ca. 10% of its length and appeared with a load exceeding 30 kN.
An example diagram of the load-strain relationship obtained from readings from an electrofusion strain gauge (TENMEX TFs-10, resistance 120 Ω ± 0.2%, Lodz, Poland) for different CFRP tape reinforcement locations is shown in Figure 7, respectively. The diagrams of the load-strain relationship based on the readings of the remaining strain gauges (T1 and T2) are shown in the further figures. The value of the destructive force was adopted at the level of 25 kN. This assumption is due to the fact that up to this load level the correct readings were obtained for all strain gauges. Consequently, further analyses were carried out assuming that the limit load was equal to 25 kN. Moreover, at this load level, the effect of using CFRP tapes on reducing the displacements and strains of the tested sigma beams was described. Table 2 shows the strain values of each of the tested beams, obtained on the basis of readings from electrofusion strain gauges at a load level of 25 kN. The percentage increase or reduction in the strain of each tested beam in relation to the strain determined for the reference beams was calculated based on the following formula: where: -reduction or increase in strain of a given sample expressed as a percentage, -strain of a given sample, and -arithmetic mean value of the two reference beams strain (B1R and B2R). The value of the destructive force was adopted at the level of 25 kN. This assumption is due to the fact that up to this load level the correct readings were obtained for all strain gauges. Consequently, further analyses were carried out assuming that the limit load was equal to 25 kN. Moreover, at this load level, the effect of using CFRP tapes on reducing the displacements and strains of the tested sigma beams was described. Table 2 shows the strain values of each of the tested beams, obtained on the basis of readings from electrofusion strain gauges at a load level of 25 kN. The percentage increase or reduction in the strain of each tested beam in relation to the strain determined for the reference beams was calculated based on the following formula: where: ρ εi -reduction or increase in strain of a given sample expressed as a percentage, ε i -strain of a given sample, and ε ref -arithmetic mean value of the two reference beams strain (B1R and B2R).
From the obtained results, the arithmetic mean value was determined for the beams reinforced in the upper flange (MBG) and reinforced in the bottom flange (MBD), which is presented in Figure 8. From the obtained results, the arithmetic mean value was determined for the beams reinforced in the upper flange (MBG) and reinforced in the bottom flange (MBD), which is presented in Figure  8. As already mentioned, a measurement of vertical and horizontal displacements at four selected points was performed using the Aramis system. An example of a load-displacement diagram is shown in the Figure 9. Then, using the Formula (2), the displacement percentage change in relation to the reference beams at a load of 25 kN and for different positions of the CFRP reinforcement was determined.
where: -percent change in displacement of the ith sample, -displacement of the ith sample, and -displacement of reference beam. The percent change in displacement is shown in Figure 10.  As already mentioned, a measurement of vertical and horizontal displacements at four selected points was performed using the Aramis system. An example of a load-displacement diagram is shown in the Figure 9. From the obtained results, the arithmetic mean value was determined for the beams reinforced in the upper flange (MBG) and reinforced in the bottom flange (MBD), which is presented in Figure  8. As already mentioned, a measurement of vertical and horizontal displacements at four selected points was performed using the Aramis system. An example of a load-displacement diagram is shown in the Figure 9. Then, using the Formula (2), the displacement percentage change in relation to the reference beams at a load of 25 kN and for different positions of the CFRP reinforcement was determined.
where: -percent change in displacement of the ith sample, -displacement of the ith sample, and -displacement of reference beam. The percent change in displacement is shown in Figure 10.  Then, using the Formula (2), the displacement percentage change in relation to the reference beams at a load of 25 kN and for different positions of the CFRP reinforcement was determined.
where: ρ ui -percent change in displacement of the ith sample, u i -displacement of the ith sample, and u ref -displacement of reference beam. The percent change in displacement is shown in Figure 10. It can be noticed that in the case of low beams subjected to significant rotation, bonding of the CFRP tape to the upper (compressed) flange contributed to the reduction in horizontal displacements to a lesser extent compared to the case of placing CFRP on the bottom (tensioned) flange. Moreover, in the case of displacements in the vertical direction it even contributed to the increase in displacement. This is surprising, because in the case of high beams [1], gluing the tape to the upper flange allowed for a significant limitation of horizontal displacements. In the case of the analyzed, i.e., low and slender beams, it can be stated that placing the CFRP tape on the upper flange is more advantageous when the aim is to limit the strain of the beams.

Numerical Model
The model of thin-walled steel beams made of the sigma-type cross-section (Σ140 × 70 × 2.5 profile) was made with the use of shell finite elements (FEs) with a linear shape function. The washers used to transfer the load directly from the press to the beam and the support clamps (whose shape It can be noticed that in the case of low beams subjected to significant rotation, bonding of the CFRP tape to the upper (compressed) flange contributed to the reduction in horizontal displacements to a lesser extent compared to the case of placing CFRP on the bottom (tensioned) flange. Moreover, in the case of displacements in the vertical direction it even contributed to the increase in displacement. This is surprising, because in the case of high beams [1], gluing the tape to the upper flange allowed for a significant limitation of horizontal displacements. In the case of the analyzed, i.e., low and slender beams, it can be stated that placing the CFRP tape on the upper flange is more advantageous when the aim is to limit the strain of the beams.

Numerical Model
The model of thin-walled steel beams made of the sigma-type cross-section (Σ140 × 70 × 2.5 profile) was made with the use of shell finite elements (FEs) with a linear shape function. The washers used to transfer the load directly from the press to the beam and the support clamps (whose shape reflected the fork support), similarly to the tests described in the paper [1], were modelled as non-deformable shell elements.
Both clamps and washers were assigned reference points in which the necessary boundary conditions were directly defined. Reference points have already been defined at the design stage of these parts (for washers these points are placed at the center of gravity, and support clamps are placed at the actual joints). The load was realized by applying two forces corresponding to the direction of the Y axis (each of the forces equal to 16,000 N), at reference points associated with non-deformable washers.
The numerical model takes into account the properties of the contact interaction in the normal direction (using Hard Contact) and tangential (without friction). In order to properly reproduce the experimental studies, contact interactions were introduced between the steel beam and support clamps and the beam and washers (used to transfer the load). The numerical model consisted of 16,104 nodes with a number of FEs equal to 15,587 (of which 14,147 were S4R-type elements, i.e., deformable shell elements), while 1440 were R3D4-type elements, i.e., non-deformable linear shell elements, from which support clamps and washer components were prepared. The material behavior was described by means of an elasto-plastic model with the bilinear characteristics [24,25]. In the case of reinforced beams, CFRP tapes (made of orthotropic material [26,27]) were modelled as shell FEs connected directly to the beam using TIE relations. The material properties used in numerical calculations are presented in Table 3. The boundary conditions of the numerical model are shown in Figure 11.
Materials 2020, 13, x FOR PEER REVIEW 10 of 16 reflected the fork support), similarly to the tests described in the paper [1], were modelled as nondeformable shell elements. Both clamps and washers were assigned reference points in which the necessary boundary conditions were directly defined. Reference points have already been defined at the design stage of these parts (for washers these points are placed at the center of gravity, and support clamps are placed at the actual joints). The load was realized by applying two forces corresponding to the direction of the Y axis (each of the forces equal to 16,000 N), at reference points associated with non-deformable washers.
The numerical model takes into account the properties of the contact interaction in the normal direction (using Hard Contact) and tangential (without friction). In order to properly reproduce the experimental studies, contact interactions were introduced between the steel beam and support clamps and the beam and washers (used to transfer the load). The numerical model consisted of 16,104 nodes with a number of FEs equal to 15,587 (of which 14,147 were S4R-type elements, i.e., deformable shell elements), while 1440 were R3D4-type elements, i.e., non-deformable linear shell elements, from which support clamps and washer components were prepared. The material behavior was described by means of an elasto-plastic model with the bilinear characteristics [24,25]. In the case of reinforced beams, CFRP tapes (made of orthotropic material [26,27]) were modelled as shell FEs connected directly to the beam using TIE relations. The material properties used in numerical calculations are presented in Table 3. The boundary conditions of the numerical model are shown in Figure 11.

FEM Results
During the numerical analysis, the strains of the beam were measured in places corresponding to the actual location of T1, T2 and T3 electrical resistance strain gauges used directly during laboratory tests. The strain from the Abaqus program was read as Max. In-Plane Principal (Abs).
In numerical analyses, the beam without reinforcements was marked as MBRa, while the reinforced beam in the upper flange was marked as MBGa and the beam reinforced with CFRP tape in the bottom flange was marked as MBDa. Comparison of laboratory and numerical results are shown in Figure 12.

FEM Results
During the numerical analysis, the strains of the beam were measured in places corresponding to the actual location of T1, T2 and T3 electrical resistance strain gauges used directly during laboratory tests. The strain from the Abaqus program was read as Max. In-Plane Principal (Abs).
In numerical analyses, the beam without reinforcements was marked as MBRa, while the reinforced beam in the upper flange was marked as MBGa and the beam reinforced with CFRP tape in the bottom flange was marked as MBDa. Comparison of laboratory and numerical results are shown in Figure 12.  Figure 13 shows a graph similar to Figure 8. Based on the results obtained from the numerical analysis, it can be concluded that in the case of points T1 and T3 a good agreement with the results of laboratory tests was obtained. In the case of reading at the point of the T2 strain gauge, the results of numerical analyses differ from the results of laboratory tests. In laboratory tests, placing the CFRP tape on the bottom flange allowed for the greatest reduction in deformations in the top flange, while in the case of numerical analyses the situation was the opposite. This is due to the fact that the laboratory tests were carried out on several beams, which showed a certain dispersion of the results, while the data presented in Figure 8 are the arithmetic mean of the obtained results.  Figure 13 shows a graph similar to Figure 8. Based on the results obtained from the numerical analysis, it can be concluded that in the case of points T1 and T3 a good agreement with the results of laboratory tests was obtained. In the case of reading at the point of the T2 strain gauge, the results of numerical analyses differ from the results of laboratory tests. In laboratory tests, placing the CFRP tape on the bottom flange allowed for the greatest reduction in deformations in the top flange, while in the case of numerical analyses the situation was the opposite. This is due to the fact that the laboratory tests were carried out on several beams, which showed a certain dispersion of the results, while the data presented in Figure 8 are the arithmetic mean of the obtained results. During the laboratory tests, displacements of samples were controlled at four points, marked P1, P2, P3 and P4. Figure 14 presents the results of vertical displacements for selected points to check the compatibility of numerical analyses and laboratory tests.  During the laboratory tests, displacements of samples were controlled at four points, marked P1, P2, P3 and P4. Figure 14 presents the results of vertical displacements for selected points to check the compatibility of numerical analyses and laboratory tests. During the laboratory tests, displacements of samples were controlled at four points, marked P1, P2, P3 and P4. Figure 14 presents the results of vertical displacements for selected points to check the compatibility of numerical analyses and laboratory tests.  Analyzing the graphs presented in the above figures, concerning the strains obtained for the beams, it should be stated that the numerical models, in terms of quality, represent the actual response of structures reinforced with CFRP tapes. In other words, they represent the same trend in terms of both laboratory test results and numerical analyses. In quantitative aspects, the strain readings from electric resistance strain gauges for the beams of a given group (reference, reinforced in the lower and upper flange) slightly differed. At a level of load of 25 kN, the difference between the results obtained reached a maximum of 12% and an average of 3.5-5.5% for strain gauges T2 and T3. In the case of displacement, the results of laboratory tests and numerical analyses, presented in Figure 14, are characterized by a similar character of the load-displacement equivalence path. The discrepancy between the arithmetic mean value of the displacement, obtained for beams with the same reinforcement method, and the corresponding FEM numerical model, in none of the analyzed cases, exceeded 13%. In addition, the form of beam deformation obtained on the basis of numerical and laboratory tests ( Figure 15) is compatible, which additionally confirms the proper method of modelling the tested thin-walled sigma beams. Analyzing the graphs presented in the above figures, concerning the strains obtained for the beams, it should be stated that the numerical models, in terms of quality, represent the actual response of structures reinforced with CFRP tapes. In other words, they represent the same trend in terms of both laboratory test results and numerical analyses. In quantitative aspects, the strain readings from electric resistance strain gauges for the beams of a given group (reference, reinforced in the lower and upper flange) slightly differed. At a level of load of 25 kN, the difference between the results obtained reached a maximum of 12% and an average of 3.5-5.5% for strain gauges T2 and T3. In the case of displacement, the results of laboratory tests and numerical analyses, presented in Figure 14, are characterized by a similar character of the load-displacement equivalence path. The discrepancy between the arithmetic mean value of the displacement, obtained for beams with the same reinforcement method, and the corresponding FEM numerical model, in none of the analyzed cases, exceeded 13%. In addition, the form of beam deformation obtained on the basis of numerical and laboratory tests ( Figure 15) is compatible, which additionally confirms the proper method of modelling the tested thin-walled sigma beams.

Results and Discussion
The following paper presents a method of retrofitting steel thin-walled sigma beams with CFRP tapes. As a result of the conducted research and analyses, satisfactory results were obtained, showing the effectiveness of this type of reinforcement. A good agreement was obtained between the finite element method analysis and the experimental studies.
The 140-mm high cold-formed sigma steel beams described in this article were characterized by a smaller cross-section wall slenderness and greater member slenderness ratio, referring to those tested in [1]; therefore, global forms of deformation were observed. The beams described in [1] (higher and with a thinner cross-sectional wall) were characterized by a greater cross-section wall

Results and Discussion
The following paper presents a method of retrofitting steel thin-walled sigma beams with CFRP tapes. As a result of the conducted research and analyses, satisfactory results were obtained, showing the effectiveness of this type of reinforcement. A good agreement was obtained between the finite element method analysis and the experimental studies.
The 140-mm high cold-formed sigma steel beams described in this article were characterized by a smaller cross-section wall slenderness and greater member slenderness ratio, referring to those tested in [1]; therefore, global forms of deformation were observed. The beams described in [1] (higher and with a thinner cross-sectional wall) were characterized by a greater cross-section wall slenderness and a smaller member slenderness ratio, so they were characterized by local forms of deformation. The beneficial effects of reinforcing with tapes were obtained for both high and low cross-section beams-i.e., for elements with different slenderness ratios.
The authors of this study admit that in the case of cold-formed elements, due to the small cross-sectional area, the reinforcement with CPRF tapes may itself increase the cross-sectional geometrical characteristic of the element and thus increases its bearing capacity. It is worth noting, however, that as a result of the reinforcement of the beams with a tape located at the bottom flange of the steel beam, a 22% increase in the moment of inertia about the y axis, 8% increase about the z axis and a change in the bending section modulus with respect to the z axis by 5% were obtained.
It should be noted that by gluing the CFRP tape to the bottom flange, it is possible to reduce the vertical beam displacement by 13% and the horizontal beam displacement by 34%. It should therefore be emphasized that the benefits of reinforcing the beam cross-section using CFRP tapes are much greater than would result from only the change of the geometrical characteristics of its cross-section. This confirms the high efficiency of reinforcing thin-walled steel sections with CFRP tapes. As it was already mentioned in [1], attention has to be drawn to the fact that the proposed method differs from the classic approach to reinforcing steel structures.
The proposed method deliberately deviates from the principle of convergence of the centers of gravity of both the beam cross-section and CFRP reinforcement. This is dictated both by the fact that the sigma-type section has only one symmetry axis, as well as due to the limitations related to the access to the element during the reinforcement process "in situ".
In addition, when designing the reinforcement of steel structure elements with tapes, various possible forms of failure should be taken into account, such as adhesive debonding along the surface of the steel-adhesive or composite-adhesive interface, CFRP delamination, and cohesive failure in the adhesive layer. This is due to the fact that the steel strength is higher than that of traditional adhesives used to strengthen structures [22].
In the presented stage of the research, as well as in the tests described in [1], a constant thickness of the adhesive layer was used, equal to 1.3 mm. It should be noticed that in [1], the tape was detached at the steel-glue interface in each of the tested samples. In the tests described in this work, debonding occurred in only one sample at a high level of load. This may indicate that the debonding of the tapes was related to the local deformation of the beams. The authors are still unable to indicate the relationship between the adhesive layer thickness and the value of load at which the tape detaches from the tested beams. The influence of the glue layer thickness on the effectiveness of reinforcement using CFRP tapes will constitute the next stage of laboratory research.
Due to the many possible forms of failure as well as the limited knowledge of the behavior of CFRP tapes glued to steel, it will be advisable to conduct further research before implementing the proposed method in engineering practice.
The aim of the study was to answer the question posed by the industry on the possibility of reinforcing steel cold-formed roof purlins made of sigma-type profiles to be used in steel halls. The conducted tests were of a pilot nature and were aimed at demonstrating the effect of the location of CFRP tapes on steel sigma beams in laboratory conditions. The obtained results do not refer directly to the reinforcement of the existing structure-for example, by the fact that it would have to be completely unloaded and devoid of cooperation with other elements. The satisfactory results obtained in the research concerning the limitation of displacement and strain of the tested beams strengthened with the CFRP tape are the motivation for further research on the reinforcement of roof purlins.

Conclusions
The obtained results of laboratory tests and numerical analyses allow us to state that the reinforcement in the form of CFRP tapes has a positive effect on the displacement and reduction in deformations of thin-walled cold-formed steel beams made of Σ140 × 70 × 2.5 profiles. Based on the performed analyses, detailed conclusions were formulated: • Placing the composite tape on the bottom flange of the beam allowed to reduce the strain in the bottom flange by 15%.

•
By sticking a composite tape (CFRP) to the top (compressed) flange, it is possible to reduce the average strain on the top flange (33%) and in web (52%).

•
Placing the composite tape on the bottom flange reduced the horizontal displacement perpendicular to the longitudinal axis of the beam on the top flange by 34%.

•
Placing of the composite tape on the bottom flange allowed to reduce the vertical displacement of the beam by 13%.
Summing up, it should be noted that reinforcement with CFRP tape adhered to the upper flange can be very advantageous for beams subjected to high rotation, when the aim is to limit the strains of the beams. The innovative approach presented in the paper, which consists of placing the tape on the inside of the beam flange, allows to remove the technological limitations of reinforcing beams during exploitation. Moreover, the authors, as in [1], performed displacement measurements using an innovative method based on the use of the Tritop system in combination with two Aramis system devices, which recorded the behavior of the tested beam on both sides. This method allows for a reliable measurement of the displacements of points subject to large displacements, which is impossible in the case of traditional methods-e.g., when using inductive displacement gauges. Overall, it can be concluded that the traditional approach to the strengthening of steel beams, according to which it is recommended to place CFRP tapes at the bottom flange, cannot be considered as a universal method. In the case of low beams, which undergo global deformation during bending, the placement of CFRP tapes at the tensioned flange allowed the limiting of displacements to the greatest extent, but it did not prove to be the best solution in the case of the need to limit the deformations of the tested beams.