The State of Deformation and Stiffness Analysis of RC Beams Strengthened by Means of CFRP Strips at Different Load Levels

Abstract

This work presented some selected results of laboratory tests and FEM analysis study of simple supported RC beams strengthened using carbon strips. The beams were examined to establish the effectiveness of this method of strengthening in terms of increasing their load-carrying capacity as well as flexural stiffness at different preloading states. A set of beams was divided into five groups, which differ in the level of the load applied before application of the composite strips on their bottom surfaces. Laboratory tests were supplemented with numerical analyses based on the finite-element method (FEM) using Abaqus software. The created numerical models were validated, and good agreement of the experimental results with the results obtained in the numerical analyses was observed. The deformation state existing in the main reinforcing bars, in concrete as well as in the strengthening composite strip, and its influence on the failure mechanism of the beams were analyzed. In the stiffness analysis, it was assumed that the stiffness of a beam strengthened with composite material after the elastic range (concrete and reinforcement steel) can be represented by the relation between the stiffness of the noncracked section B and beam curvature κ. The curvature–bending moment diagram as well as bending moment–stiffness diagram were prepared on the basis of laboratory and numerical results. Based on the results, it can be stated that the preload on the beam before strengthening affects the levels of deformation and the utilization rates of the composite material as well as reinforcing bars. The curvature and stiffness of the beams depend on the load level at which the CFRP strip strengthening is realized. The results of the analysis of preloaded beams before strengthening indicate that totally relieving them prior to strip application turns out to be the most beneficial solution.

1. Introduction

The method of strengthening RC (reinforced concrete) elements by bonding to their zones of tension and/or shear composite elements made of carbon fibers (CFRP—carbon-fiber-reinforced polymer or plastics), aramid fibers (AFRP—aramid-fiber-reinforced polymer or plastics), or glass fibers (GFRP—glass-fiber-reinforced polymer or plastics) is still a quite new method, which is becoming more and more popular [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. The increased popularity of CFRP materials is due to their outstanding features compared with other more traditional materials, such as light weight, linear deformation up to failure, very high tensile strength and modulus of elasticity, resistance to most corrosive agents, very high resistance to fatigue, and easy adaptation of composite element geometries to the shape of strengthened RC elements, etc. In addition, although the issues related to composite materials are still researched and analyzed in order to have a better and more effective use of their advantages, many researchers in their work summarized the current state of the art [6,7,8,9,10,11,12]. This often helps to find areas of knowledge that are still not fully explored.

The use of carbon strips on the bottom surfaces of reinforced concrete beams increases not only their load-carrying capacity but also their stiffness. The essence and scope of stiffness changes are important both in the serviceability limit states (change in deflections) and, which seems to be more important, in the ultimate limit states of statically indeterminate systems. The distribution of internal forces in concrete elements is determined by the stiffness of the cracked cross-section, whereas changes in the stiffness in the structure with changing load determine the redistribution of internal forces. Thus, it is not without significance in what places, with what cross-section, and at what level of load on the structure the carbon strip will be applied or glued to the concrete surface. The level of the beam load at which we strengthen the beam is important because the properties of concrete, especially in tension, are significantly dependent on the state of their stress. Exceeding the limit deformation of the concrete in tension leads to irreversible changes in the structure of the concrete, which significantly affects the stiffness of the reinforced cross-section depending on the stage in which it is strengthened. By appropriately controlling the method of strengthening the structure, we can actively influence not only the deflection values but also the increase in the load capacity of the structure using the same amount of added material.

The issues of strengthening preloaded and precracked reinforced concrete beams are nowadays considered by many scientists and research teams [3,4,13,14,15,16,17,18,19,20,21,22]. Laboratory tests are very often organized, supported by nonlinear numerical analysis (after positive verification of the created models and adopted modeling rules), which provide the most complete information about the behavior of the RC beams initially cracked before the application of CFRP strip strengthening. This complex approach and results of investigation on the influence of initial loading and existence of cracks prior to rehabilitation on the CFRP strengthening effect were presented in [19,20,21,23,24,25]. An example of an efficient usage of numerical analyses in parametric analyses (carried out on experimentally validated models and numerical procedures for a passive way of CFRP strengthening) is presented in [26], where the active strengthening of RC elements with pre-tensioned strips was carefully investigated.

Scientists have noticed not only the increase in the load capacity of elements strengthened with CFRP strips. The analysis of flexural stiffness is also undertaken in scientific works. In [27], the researchers tried to assess the effectiveness of CFRP-repaired RC beams under different damage levels based on flexural stiffness. The analysis of the stiffness of reinforced concrete beams with CFRP strips also took place in [3,4]. Flexural stiffness is important not only in RC elements reinforced with CFRP strips but also in other structural elements, e.g., wooden beams strengthened using carbon or glass fiber materials. It was confirmed by Nadir et al.’s team in [28].

This work is another contribution to the research on the strengthening with CFRP strips of the initial loaded reinforced concrete beams, with particular attention to the analysis of the state of deformation and stiffness of strengthened RC elements. Presented results of the laboratory tests as well as results of numerical analyses by means of the FEM aimed to show the impact of the initial deformation state and the current load level at which the strips were applied to beam surfaces on the deformation state, failure mechanism, and stiffness of reinforced concrete elements. The high importance of the undertaken issues is due to the fact that in building structures, the safe operation of reinforced concrete elements is determined by not only their work in the range of loads correspondingly lower than the limit load capacity but also the stiffness of the element throughout the entire operating range.

2. Scope of Analysis

Five series of two beams were tested, differing in the level of the load at which the strengthening with the CFRP strip was implemented [3,4]. All beams of span length 3.0 m were subjected to four-point bending (Figure 1 and Figure 2). The load program was assumed to reflect the nature of work of beams in bridge structures. The series of tested beams differed from each other by the level of applied loads before the CFRP strip application. The load levels have been established in relation to the design capacity of an RC beam without strengthening. The following series of beams were tested:

• Beam type Br—reference RC beams with no CFRP strip; used as control specimens;
• Beam type Bz—RC beams "reinforced" with CFRP strips before the beginning of load application;
• Beam type Boow—RC beams with CFRP strips, examined in two stages (stage 1—RC beam loaded to P = 60 kN, equal to the level of the design load-carrying capacity of reinforced concrete beam without strengthening (which corresponds to ~62% of the load-carrying capacity of such beam), then totally relieved; stage 2—beam strengthened by means of CFRP strips and then loaded up to failure);
• Beam type Bow—RC beams with CFRP strips, examined in two stages (stage 1—RC beam loaded to P = 60 kN, then relieved to P = 45 kN, equal to ~75% of the design load-carrying capacity of the beam without strengthening (it corresponded to the dead weight of a bridge structure); stage 2—beam strengthened using CFRP strips at a load level of P = 45 kN and then loaded up to failure;
• Beam type Bopw—RC beams with CFRP strips, examined in two stages (stage 1—RC beams loaded to P = 75 kN, equal to a value of 25% above the design load-carrying capacity of the RC beam without strengthening, then relieved to P = 45 kN, equal to ~75% of the design load-carrying capacity of the beam without strengthening; stage 2—beam strengthened using CFRP strips at a load level of P = 45 kN and then loaded up to failure).

The side view and cross-section of all tested beams together with the layout of reinforcing bars are shown in Figure 1. On the bottom surfaces of all strengthened beams, CFRP strips were applied due to flexure (S&P Lamelle CFK type 150/2000). Each composite strip had a length of 2.80 m and cross-section of 0.05 m × 1.2 mm and was bonded to concrete surfaces by means of a two-component epoxy resin adhesive [3]. A detailed description of the test stand, used materials, test procedure, and main results were presented in [3,4].

At the beginning, comprehensive measurements of all used materials have been conducted. The mean measured yield strength and modulus of elasticity of the 12 mm reinforcing bars were 550.80 MPa and 210.50 GPa, respectively. The 4.5 mm reinforcing bars, without explicit yield plateau, had a 0.2% yield proof stress of 554.70 MPa and a modulus of elasticity of 193.30 GPa. Poisson’s ratio for both bar types was assumed as 0.3. The modulus of elasticity and Poisson’s ratio of the CFRP strips were measured as 158.95 GPa and 0.2, respectively.

All analyzed beams have been made of concrete of strength class C40/50, according to CEB-FIP Model Code 1990 [29]. The maximum aggregate size (gravel) was specified as 16 mm. The concrete cube, cylinder, and small concrete beam (cuboid) specimens were made at the time of casting and were kept in the same condition as the beams during curing. The modulus of elasticity and Poisson’s ratio before cracking determined on cylinder specimens were measured as 29.98 GPa and 0.164, respectively. The concrete compressive cube strength was 54.00 MPa, and the concrete compressive cylinder strength was 44.40 MPa. The compressive stress–strain relationship was adopted as a parabola diagram according to Rüsch [30]. The tensile strength of concrete was determined from the indirect tests (in flexural and splitting cylinder tests) and was equal to 3.467 MPa.

Based on the material laboratory tests of reinforcing steel, composite strip, and concrete (by means of the direct and indirect methods), accurate material characteristics were defined. Then the load-carrying capacity of the reference RC beam cross-section with no CFRP strip was calculated and then assumed as M_R = 47.0 kNm. It corresponded to the ultimate load equal to 2 · P = 94.0 kN. The test stand was located under the frame of the hydraulic testing machine. The view on the research stand with a sample during examination is shown in Figure 2.

During the experiments, deflections of beams, strains of concrete in the midspan of beams, strains along the middle reinforcing bar, as well as the longitudinal strains along the composite strip were recorded. The force applied to the beam was also controlled and recorded. Furthermore, constant monitoring of crack arrangement on the side and bottom surfaces of all beams at each level of loading was performed. The arrangement of measurement points on all beams is shown in Figure 3.

In the experiment, failure of all strengthened beams rapidly occurred without being previously signaled by large deflections or large opening of the main cracks. The destruction was accompanied by the crackle of the CFRP strip detached from the surface of the concrete and the crackle of the cracking of the concrete.

3. Nonlinear Finite-Element-Method Analysis

In the previous author’s study, finite-element-method (FEM) models of RC beams were created [3,4], and numerical calculations were carried out in the environment of Abaqus Standard software. In order to describe the behavior of concrete, the concrete damage plasticity (CDP) material model was used [31,32,33,34]. It is a continuum, plasticity-based, damage model for concrete. It can be used for plain concrete, even though it is primarily intended for the analysis of RC structures. It was assumed that reinforced concrete can work in tensile zones even after cracks occur ("tension stiffening" effect).

In order to represent the behavior of concrete after cracking, the description of concrete expressed by a function of the fracture energy G_f released in the process of crack formation was used [3,4]. The strain-softening behavior of concrete in tension was given as the stress–displacement σ–w (stress–width of crack) according to the idea of a fictitious crack model [35]. Tension-stiffening effect was presented by applying the fracture energy cracking criterion. Such an approach enabled to account for the effects of reinforcement interaction with concrete to be simulated in a simple manner. In the numerical analysis, the four-parameter model was used [36] as one of the possible simplifications of the relationship σ–w (stress–width of crack) received during the experiments [37]. The reinforcing bars were modeled in a discrete manner as truss elements embedded in 3D or 2D solid elements of concrete and were treated similar to the linear elastic–plastic material with isotropic hardening. The presence of an adhesive between the RC beam surface and CFRP strips was neglected. It was assumed that all strengthening elements were perfectly bonded to concrete surfaces.

Figure 4a shows the overall layout of the 2D model of a half of the beam with an indication of the method of supporting and applying the load as well as the method of connection between the nodes of the beam and nodes of the strip. The finite-element model of RC beams used in computations is shown in Figure 5b. In the model, the symmetry of the beams was taken into account. Hence, the model of a half beam was made with the appropriate displacement constraints imposed on the appropriate axis of symmetry.

All numerical analyses were performed according to the earlier accepted loads and to the same program of strengthening, which had been carried out in the experiments. It is worth to mention that, in the numerical analyses, the ultimate forces of all beams were assumed using the strain criterion in steel rebar in the pure bending zone equal to two times the yield strain equal to 0.005.

4. Results

This section presents the results of all analyses carried out on the basis of the results of laboratory tests and numerical calculations. The comparison of the beam load path obtained from the experiments with the results of numerical analyses will confirm the correctness of the proposed numerical model. Next, the deformation state of reinforcing steel and composite strips of all beams strengthened with CFRP strips under changing load will be discussed. Finally, an analysis of the flexural stiffness of beams strengthened with composite strips will be carried out on the basis of the results of laboratory tests and numerical calculations.

4.1. Load-Carrying-Capacity Analysis; Comparison of Experiment and FEM Analysis Results

The extensive analysis of failure mechanisms of all individual beams was performed based on the character of their failure and crack layouts. Both beams with no composite strip (type Br) were destroyed by over-yielding of the reinforcing bar, which had occurred before the concrete was crushed in the compression zone in the middle of the beam span. All the strengthened beams were damaged by the delamination of the composite material from the concrete surface, which was always brittle.

In

Table 1. Values of ultimate load and maximum deflections during failure for laboratory-tested beams.

No.	Beam	Ultimate Load [kN]	Mean Ultimate Load [kN]	Max. Deflection at Failure [cm]	Mean max. Deflection at Failure [cm]
1	Br-2	97.13	96.48	4.06	4.19
2	Br-3	95.85		4.32
3	Bz-1	124.25	125.41	2.78	2.81
4	Bz-2	126.57		2.83
5	Boow-2	118.59	119.55	2.60 1	2.64 1
6	Boow-3	120.50		2.68 1
7	Bow-1	116.39	115.62	2.90	2.83
8	Bow-2	114.84		2.76
9	Bopw-1	117.39	116.71	2.84	2.88
10	Bopw-2	116.02		2.91

¹—Values gained in stage 2, disregarding permanent residual deflections remaining at the end of stage 1.

, the values of ultimate forces and maximum deflections of all laboratory-tested beams at the moment of their failure are put together, whereas their relative values are given in

Table 2. Relative values of ultimate load and maximum deflections during failure for laboratory-tested beams.

No.	Beam	Load-Carrying Capacity [-]	Max. Deflection at Failure [-]
1	Br	1	1
2	Bz	1.30	0.67
3	Boow	1.24	0.63 1
4	Bow	1.20	0.68
5	Bopw	1.21	0.69

¹—Values gained in stage 2, disregarding permanent residual deflections remaining at the end of stage 1.

. Examples of the deflections of the B-4 point in the middle of the beam span (according to Figure 3) received in experiments are shown in Figure 5a. In Figure 5b, the positive comparison of the FEA and experiment results is also presented—deflection of point B-4 for the RC beam (Br) and for the beam with a CFRP strip (Bz). The high compatibility of the results visible in the chart confirms the correctness of the preparation of the FEM model and the entire procedure of numerical calculations.

The comparison of the ultimate loads for all beams received in the numerical analyses and recorded during laboratory studies is summarized in

Table 3. Ultimate load received using FEM analysis and experiments [kN].

	Br	Bz	Boow	Bow	Bopw
Results of FEM	95.00	124.00	118.00	116.00	115.00
Results of experiments	96.48	125.40	119.60	115.60	116.70

. It is worth mentioning that the ultimate load of the RC beam without a CFRP strip, calculated on the basis of the results of the material tests, was 94.00 kN.

4.2. State of Deformation Analyses of Reinforced Concrete Beams Strengthened with CFRP Strips

In the experiments, deformations were measured at selected measurement points located on the composite strip, middle reinforcing bar, and top and side surfaces of the concrete in the compressed zone in the middle of the beam span (Figure 3).

The values of strains of composite strips and reinforcing bars in the middle of the beam span for different load levels are presented in

Table 4. Values of strains of CFRP strip and the bar in the middle of beam span.

No.	Beam	Max. Strain ε11 at Failure [‰]		ε11strip/ε11bar	Strain ε11 for Load P = 60 kN in the Middle of Beam [‰]		ε11strip/ε11bar
		CFRP Strip	Bar		CFRP Strip	Bar
1	Br-2	x	17.36	x	x	1.87	x
2	Br-3	x	67.87 2	x	x	2.18	x
3	Bz-1	6.07	14.38	0.42	1.66	1.79	0.93
4	Bz-2	6.45	35.45	0.18	1.67	1.82	0.92
5	Boow-2	5.19	13.89 1	0.37	1.43	1.90 1	0.75
6	Boow-3	5.98	4.57 1	1.31	1.46	2.21 1	0.66
7	Bow-1	4.82	12.32	0.39	0.38	1.95	0.19
8	Bow-2	4.58	16.02	0.29	0.38	2.00	0.19
9	Bopw-1	5.32	32.94	0.16	0.38	2.08	0.18
10	Bopw-2	5.06	9.83	0.51	0.28	2.12	0.13

¹—Values taking into account permanent strains remaining after the beam is loaded before it is strengthened with the strip. ²—Steel strain of reinforcing bar in the middle of the beam span after yielding.

. Table 4 is illustrated in Figure 6 and Figure 7, which show the quantitative relationship between the strip and reinforcing steel strains for individual beams for the load P = 60 kN and for the beam failure moment. Table 4 shows that the initiation of the destruction of all beams took place in the reinforcing bars in the central zone of beams due to the steel yielding (ε_pl = 2.5‰). The rapid increase in the deformation of reinforcing bars on the central sections of the beams resulted in an immediate increase in the deformation of composite strips on these sections. The maximum strains of the composite strip with the failure of the beams ranged from 4.58‰ for the beam loaded before the strengthening with the strip (Bow type) to 6.45‰ for the beam strengthened with the strip from the beginning of loading (Bz type).

The high utilization of the composite strip at the level of 5.98% was obtained for the beam completely relieved before strengthening. On the other hand, the results of the analysis of CFRP strip strains of beams that were strengthened at the load corresponding to 75% of the design capacity of the reinforced concrete beam (Bow and Bopw beams) show that the greater the overload of the beams before strengthening, the greater the use of the CFRP strip strength at the time of their failure.

The load levels at which the reinforcing steel reaches the yield limit are summarized for all beams in

Table 5. Determination of the beginning of steel yielding of the main reinforcing bar.

No.	Beam	Load of Beam at the Yield of Reinforcing Steel [kN]	Mean Value [kN]	Mean Relative Load at the Yield of Reinforcing Steel [-]
1	Br-2	84.3	83.1	1
2	Br-3	81.9
3	Bz-1	98.6	99.5	1.20
4	Bz-2	100.4
5	Boow-2	98.8	97.0	1.17
6	Boow-3	95.1
7	Bow-1	96.3	95.3	1.15
8	Bow-2	94.3
9	Bopw-1	86.7	89.9	1.08
10	Bopw-2	93.1

. The table also gives the relative values of loads causing the reinforcing steel yielding in the beams strengthened with the CFRP strips in relation to the load at which the steel was yielding in the beams with no strengthening (the value of 1 for beams without any strengthening is the reference level).

Increasing the relative levels of loads causing the steel yielding in beams strengthened with CFRP strips is one of the most important criteria for assessing the effectiveness of RC beams with composite strips. This criterion determines the range of safe operation of RC beams, which would be damaged without the CFRP strengthening as a result of exceeding the yield limit of the steel in tension. An example of steel strains in the middle reinforcing bar in the midspan of all beams (point St-3) is shown in Figure 8a.

The concrete strain diagrams on the top surface of the beams in the middle of their spans (point Bt-g as a mean value of strains measured in points Bt-3 and Bt-4 according to Figure 3) are shown in Figure 8b, and the values of the concrete strain for different work phases are summarized in

Table 6. Concrete strain values on the top surface of beams in the middle of their span.

No.	Beam	Max. Concrete Strain ε11 at Failure [‰]			Concrete Strain ε11 for Load P = 60 kN in the Middle of Beam [‰]
		Absolute Values		Relative Values	Absolute Values		Relative Values
		[‰]	Mean [‰]	[-]	[‰]	Mean [‰]	[-]
1	Br-2	2.33	2.94	1	0.77	0.84	1
2	Br-3	3.54			0.90
3	Bz-1	2.29	2.32	0.79	0.82	0.78	0.93
4	Bz-2	2.34			0.74
5	Boow-2	2.45	2.29	0.78	0.95	0.89	1.06
6	Boow-3	2.13			0.83
7	Bow-1	2.16	2.25	0.76	0.89	0.91	1.08
8	Bow-2	2.34			0.93
9	Bopw-1	2.13	2.17	0.74	0.87	0.87	1.04
10	Bopw-2	2.20			0.87

. The summary shows that, only in the case of one of the reference beams (without strengthening), reaching the ultimate limit state was accompanied by exceeding the load capacity of the compression zone of concrete (ε_gr = 2.6‰). In the remaining beams, concrete did not crush before the beams reached the limit load capacity, which confirms the correctness of the selection of the strengthening system. The maximum concrete strain due to the failure of the beams ranges, on average, from 2.17‰ in the case of beams overloaded before the strengthening with a CFRP strip (Bopw type) to 2.94‰ in the case of beams without CFRP-strip strengthening (Br type).

Additionally, the analysis covered the development of deformation of the CFRP strip and the reinforcing bar along the length of the beams for different load levels. Particular attention was paid to the mutual relations between the strip and bar strains in the same cross-sections of beams in the states preceding their failure. This was carried out for establishing the immediate cause of the initiation of failure. The layout of measurement points for recording longitudinal strains along the beams on the strip and on the middle reinforcing bar is shown in Figure 9.

When analyzing the deformation of the strips in the Bz-1 and Bz-2 beams, it can be noticed that the strips deformed approximately the same from the side of both ends of the beams, also at higher loads (Figure 9). In the Bz-1 beam, only at the loads just before its failure, at which there was a rapid increase in the strains on the reinforcing bar on one side of the beam, the strains of the strip began to increase faster toward one of the supports. This proves that, on the still extending section of the beam, the strip acted like a bowstring, only locally connected with the beam. In both Bz beams, the initiation of destruction took place in the zone of the constant bending moment of the beams, in which cracks began to intensively open due to the rapidly increasing strains of the reinforcing steel. The failure mechanism can, therefore, be described as “failure due to cracks due to bending”.

In the case of Bow and Bopw beams, but also in Boow beams, a similar failure mechanism was noticed. However, the rapid growth of steel strains of the bars on one side of the beam and taking over of some functions of the bars (after yielding of steel) by the CFRP strip were more visible. The increase in the strains of the steel bars was accompanied by breaking the adhesion between the strip and concrete surface of the beam, which developed toward one of the support points. This is confirmed by the charts presented in Figure 10 and Figure 11.

4.3. Analysis of the Stiffness of RC Beams Strengthened with CFRP Strips

The analysis of the stiffness of reinforced concrete beams is a complex issue due to the nature of the reinforced concrete cross-section, made of two materials with different strength properties. One of the methods for assessing the stiffness of an RC beam is the analysis of the section stiffness based on the M–κ relation (bending moment–curvature). Using the M–κ relationship instead of the layered model also allows you to shorten complex calculations in nonlinear analysis, such as determining the position of the neutral axis or changes in elastic stiffness (due to the elastic degradation of the material). The analysis of the M–κ relation in the RC beam was devoted to [38], and its adaptations to the elements strengthened in bending with composite materials were presented in [2,39] (analysis of the relation M–κ for elements strengthened with composite bars [40]).

A typical M–κ relationship for a reinforced concrete beam consists of three ranges (Figure 12): I—range up to cracking, II—range up to yielding of reinforcing steel, and III—range until reaching the ultimate strains in the concrete. It is clearly influenced by the dimensions of the concrete cross-section as well as the material properties of concrete and reinforcing steel (B = EI). When describing the theoretical M–κ curve for a beam subjected to bending, the rule of flat sections applies (i.e., longitudinal deformations are directly proportional to the distance from the neutral axis). A detailed description of the M–κ relationship (exactly M-1/r_i, where i is another range of work of the reinforced concrete element according to Figure 12) for pure bending and compression bending cases can be found in [29].

A similar analysis of stiffness changes can be performed for bending elements strengthened with composite materials, regardless of the place and method of their application on the beam. Changes in the stiffness of a cross-section are a function of the beam deflection in a given cross-section, i.e., also a function of curvature in this cross-section. Until the element is cracked (the value of the moment causing cracking M_cr is reached), the M–κ relationship in the strengthened beam practically corresponds to the analogous section in the unstrengthened beam. The stiffness in the ranges after cracking of the strengthened beam (ranges II and III) is influenced by the cracks (or cracks already existing before the strengthening application), the interaction of reinforcing bars with concrete, concrete crushing, and, most of all, the interaction of beam with the composite strip applied on it.

It was assumed that the stiffness of a beam strengthened with the composite material after the elastic range (of concrete and reinforcement steel) can be represented by the relation between the stiffness of the noncracked section B and beam curvature κ [3]:

$$\mathrm{B}=\frac{\mathrm{M}}{\mathsf{\kappa }},$$

(1)

where B is the stiffness of the beam section, M is the bending moment, and κ is the beam curvature.

The stiffness of the beams was calculated as a function of the curvature determined on the basis of the measured deflections of points on the top surfaces of the tested beams (points B-1 to B-7 in Figure 3).

Based on the deflection lines of the beams described by the fourth-degree function, the curvatures of points in the middle of each beam span were calculated. Then, for the known values of curvatures, the actual stiffnesses of the beams were calculated according to formula (1), in which κ means the curvature of the beam for the section in the middle of its span (between the applied forces), and M is the current bending moment in this section.

Figure 13 presents the curvature and stiffness diagrams both depending on the bending moment for the tested beams. These plots show the curvature changes for the sections in the middle of the spans of these beams (Figure 13a). For these cross-sections, the stiffnesses were then calculated and shown in Figure 13b.

Boow-type beams (relieved before the moment of their strengthening using CFRP strips) always represent two curves in the drawings. The first one presents the characteristics of the beams taking into account their initial deflections arising after stage 1 of the test (after the loading and relieving process before the strip application), whereas the second one neglects the initial state, assuming the end of stage 1 of the tests as the beginning of the analyses, i.e., the moment when the beam is already strengthened with the strip.

For strengthened elements that are part of the structures in operation, the stiffness changes obtained in the load ranges corresponding to the actual operation of these structures are of the greatest importance. A structure without strengthening is assumed to safely operate up to a load level that corresponds to its design capacity. Therefore, this load range has been analyzed in more detail.

Table 7. Values of curvature and stiffness of CFRP strengthened beams at load of 60 kN (which corresponds to design load-carrying capacity of reference beam) based on experimental results.

No.	Beam Type	Curvature κ [1/m]	Stiffness B [kNm2]
1	Br	0.008125	3692.4
2	Bz	0.007812	3840.7
3	Boow—Stage 1 + 2	0.008184 1	3666.2
	Boow—Stage 2	0.006692 2	4483.6
4	Bow	0.008659	3464.7
5	Bopw	0.009002	3333.9

¹—Based on deflection taking into account permanent deflection remaining after stage 1. ²—Based on deflections gained in stage 2, disregarding permanent deflections remaining at the end of stage 1.

lists the curvature and stiffness of the beams for the load P = 60 kN, which corresponds to the design load capacity of a reinforced concrete beam without any strengthening (Br type).

Table 8. Relative curvature and relative stiffness of beams strengthened using CFRP strips in comparison with reference beam at load of 60 kN (which corresponds to design load-carrying capacity of reference beam) based on experimental results.

No.	Beam Type	Curvature κ [-]	Stiffness B [-]
1	Br	1	1
2	Bz	0.961	1.040
3	Boow—Stage 1 + 2	1.007 1	0.993
	Boow—Stage 2	0.824 2	1.214
4	Bow	1.066	0.938
5	Bopw	1.108	0.903

¹—Based on deflection taking into account permanent deflection remaining after stage 1. ²—Based on deflections gained in stage 2, disregarding permanent deflections remaining at the end of stage 1.

summarizes the relative curvature reductions and relative stiffness of beams strengthened with the strip compared with the curvature and stiffness of the beams without strengthening (the value of 1 for the beam with no strengthening is some reference level for the analyzed parameters of beams strengthened with the strip). The summary shows that for the load level that corresponds to the design load-carrying capacity of the reference beam with no CFRP strips, the most efficient strengthening is obtained for a fully relieved beam before strengthening with the CFRP strips. For Boow-type beams analyzed from the moment the strip is glued, the greatest increases in stiffness and the greatest reduction in curvature values are obtained.

For all strengthened beams, the presence of the CFRP strip begins to play a decisive role in increasing the stiffness of the beams only from the load level at which the main reinforcement in the beam without the strip becomes plasticized (~M = 40 kNm).

Figure 14 shows the dependence of the beam curvatures on the bending moments, calculated on the basis of the beam deflection lines obtained in the FEM analyses. It is worth noting that the beam stiffness, calculated on the basis of numerical analyses, qualitatively confirms the results obtained in laboratory tests, especially when the analysis covers the entire load range of the beams. Nevertheless, the results of the quantitative analysis (summarized in

Table 9. Stiffness and relative stiffness of beams strengthened using CFRP strips in comparison with reference beam at load of 60 kN (which corresponds to design load-carrying capacity of reference beam) based on experimental and numerical results.

No.	Beam type	Experiment		FEM Analysis
		Stiffness B [kNm2]	Relative Stiffness B [-]	Stiffness B [kNm2]	Relative Stiffness B [-]
1	Br	3692.4	1	2746.9	1
2	Bz	3840.7	1.04	3236.8	1.18
3	Boow—Stage 1 + 2	3666.2 1	0.99	2875.3 1	1.05
	Boow—Stage 2	4483.6 2	1.21	-	-
4	Bow	3464.7	0.94	2732.6	0.99
5	Bopw	3333.9	0.90	2233.8	0.81

¹—Based on deflection taking into account permanent deflection remaining after stage 1. ²—Based on deflections gained in stage 2, disregarding permanent deflections remaining at the end of stage 1.

) show a large discrepancy between the stiffness values obtained in the experiment and numerical analyses, especially when it relates to a specific value of the applied load.

5. Conclusions

The presented study has confirmed that the initial loading has a significant influence on load-carrying capacity as well as deformability and stiffness of beams strengthened using composite materials. The initial state of RC beams should not be overlooked when analyzing the effectiveness of this type of strengthening.

The observations and results of experiments and FEA results became the basis for the formulation of the following general conclusions:

• The increase in the load-carrying capacity of strengthened beams without preloading in comparison with reference beams with no CFRP strip was 30%, and 24% for beams relieved before strengthening to the level of their deadweight.
• The reductions in the deflection at failure in relation to the deflection of the RC beam with no strip were 33% for beams "reinforced" with CFRP strips before the beginning of load application, 37% for fully relieved beams (neglecting the initial deflection of the beam), and 32% and 31%, respectively, for initially preloaded beams that were previously loaded to the design load-carrying capacity level of a beam without strengthening and overloaded by 25% in relation to the design load-carrying capacity of a beam with no CFRP strip.
• The preload on the beam before strengthening affects the utilization rate of the composite material. The maximum deformation of the composite strip during the failure of the beams ranged from 4.58‰ for the beam loaded before strengthening to the level of the design load-carrying capacity of the reference beam to 6.45‰ for the beam unloaded before strengthening.
• The beginning of the yielding process of the reinforcing steel for strengthened beams was observed later than for beams without strengthening. The yielding of the reinforcing steel for unloaded beams before strengthening occurred for a load 20% higher than for beams without strengthening, whereas for beams overloaded before strengthening by 25% above the design capacity, it was only 8%.
• The curvature and stiffness of the beams depend on the load level at which the CFRP strip strengthening is realized. Strengthening of unloaded beams has a significant impact on the increase in the stiffness of this beam only from the design load capacity of the beam without the strip. From this level, a much smaller increase in the curvature of the strengthened beam is visible in relation to the beam without strengthening and a significant increase in its stiffness.
• Failure of the tested beams was rapid and was not signaled early enough. The destruction was preceded by the delamination of the strip from the concrete surface. Based on the state of deformation of all beams, the initiation of failure took place in the zone of the constant bending moment, in which cracks began to intensively open due to the rapidly increasing strains of the reinforcing steel.
• Strengthening structures with composite materials is particularly effective in the case of structures that were overloaded during their operation. On the example of the analysis of the stiffness and load-carrying capacity of beams overloaded before strengthening, it can be concluded that strengthening with composite strips can be an effective treatment for increasing the load class of bridges related to road repairs and modernization, as well as it can be a solution to the problem of the safety of operation of structures at risk of unforeseen, oversized loads.
• The results of the analysis of deformation of composite strips during laboratory tests showed that, in the case of passive strengthening of RC beams with carbon strips, it is not possible to fully use their potential before the beam failure. Increasing the level of strip deformation may occur only when using an active way of strengthening, i.e., with initial pre-tension of the CFRP strip before sticking it to the concrete surface [14].
• The results of the analysis of preloaded beams before strengthening indicate that totally relieving them prior to strip application turned out to be the most beneficial. A more favorable strengthening effect is obtained if, in the analysis, we ignore the initial deflections and deformations occurring in the steps preceding the strengthening of beams. This approach best corresponds to the cases of actual strengthening of reinforced concrete structures (usually, the previously created permanent deflection of the beam is not taken into account); therefore, it is the most reliable in determining the effectiveness of strengthening the beams with a composite strip.
• The finite-element method (FEM) proved to be useful in the analysis of RC beams strengthened with CFRP elements. Validation of the numerical model and the whole procedure of numerical modeling based on the results of experiments positively ended. This proves that numerical analyses carried out on positively verified numerical models can, in many cases, replace usually costly and long-term studies on samples on a real scale.
• Summarizing the above given conclusions, it can be firmly stated that the effect of strengthening reinforced concrete beams with carbon strips not only is determined by the amount of added strengthening material and its location but also largely depends on the deformation state and crack layout of the RC element being strengthened. This applies both to the analysis of the load-carrying capacity of the RC beam strengthened using CFRP strips (in the ultimate limit states) and to the analysis of the deformation level and stiffness of such beam (in the serviceability limit states).