Torsional behavior of RC beams strengthened with carbon fiber reinforced polymer composites

Abstract

This paper shows a theoretical model for reinforced concrete beams subjected to torsion and strengthened with composite of carbon fibers (CFRP). It is based on researches of Michael Collins and collaborators and approaches the theory of diagonal compressive field. This model was inserted in a mathematic software, due to the need of interactions to obtain the torsion and torsion angles. The validation of theoretical values was realized through the results of tests obtained from the literature and it was understood that the model is valid and conservative for the ultimate torsion in most of the studies carried out. However, the beams that had the geometric ratios width and height b/h < 0.5 and the ratio between the cover and the smaller dimension of the d/cob element ≤6 had not so accurate results.

Keywords

carbon fiber composite generalized diagonal compression field theory torsion design

Introduction

Today the building industry, universities and researchers are being challenged to make new developments by means of improved technologies and methodologies that make maximum use of the available resources and minimize the cost of projects. Carbon fiber reinforced strengthening of concrete has been widely used during the last years in several countries and can be considered a well-established construction procedure. During the last few years some important contributions by researchers have introduced new design models into this reinforcing system.

This is an extremely flexible technique of strengthening structures, enabling adaptation of the material to different needs in accordance with the given architectural, structural and economic requirements. The technique of strengthening reinforced concrete elements with external bonded composite material is both economic and effective. The composite materials technology offers a wide range of different products, which vary in type, diameter and fiber orientation, and in their epoxy properties. The arrangement of the strengthening is another fundamental parameter that influences beam resistance.

The reinforcement concept to increase the performance of reinforced concrete beams submitted to torsion is not very widespread. Some studies on the torsional behavior of concrete beams reinforced with carbon fiber composites (CFRP) have been recently developed, however they remain inconclusive. The main objective of this study is to examine the parameters that influence the ultimate torsion of reinforced concrete beams strengthened with externally bonded CFRP. Drawing from existing experimental results a new approach for the ultimate torsion is investigated herein.

From the 1990s the study of reinforcement with externally bonded fiber composites on reinforced concrete elements has increased. The investigations, in general, were aimed at analyzing the strength increase of the beams bending or shear (Panchacharam and Belarbi, 2002).

In the early 2000s, were started some tests with beams submitted to torsion with Ghobarah et al. (2002), Panchacharam and Belarbi (2002), and Zhang et al. (2001). Throughout the decades of 2000 and 2010 some studies have revealed experimental results and analytical models, in which the behavior of this type of reinforcement was analyzed, allowing a better understanding of the interaction between the three materials: concrete, steel and carbon fiber composites. These studies differ in several assumptions adopted, such as: theoretical models for the torsion of the reinforced concrete beam, different orientations of the fibers (90°, 45°, or 0°), stirrups spacing, CFRP confinement effect and number of layers involving the section.

This paper proposes a theoretical model for the analysis of strengthened reinforced concrete beams with CFRP subjected to torsion, which is compared with experimental results obtained in the literature (Silva, 2017).

Theoretical models

The Federation Internacionale du Beton furnish empiric expression for strain that solve the ultimate torsion of reinforced beams strengthened with CFRP when completely involved or in U format (lateral and bottom faces), and the CFRP maximum strain for only 90° orientated fibers with the longitudinal beam direction (Fib-Bulletin 14, 2001). Stands out the publications (Ameli et al., 2004; Ghobarah et al., 2012; Mohammadizadeh et al., 2009; Mohamed and Benmokrane, 2016; Silva Filho, 2007; Tudu, 2012), for beams tests of quadrangular and rectangular sections, which are the most used in reinforced concrete structures. However, in the sizing can be considered the contribution of the slabs, thus it has beams with T sections. Chalioris (2006b), Deifalla and Ghobarah (2010), Abduljalil and Sarsam (2012) and Sure (2013) presented their studies for this case. Also tests of beams of box section were realized (He et al., 2014; Hii and Al-Mahaidi, 2006; and format L Deifalla et al., 2013).

With the aforementioned publications it is concluded that the 90° directed reinforcement, which involves the entire section along the element, is more effective than in the case of 0° and for reinforcement in U-shape. The fibers at 45° are more efficient in increasing the strength of the beam when compared to those oriented at 90°. The researchers have been increasingly validating theoretical models through computational modeling (Finite Element Method − MEF), as examples are the researches of Ameli et al. (2004), Ganganagoudar et al. (2016), Neale et al. (2005) and Tudu (2012).

Mitchell and Collins (1974) presented the Generalized Diagonal Compression Field Theory (GDCFT) model for torsion, which was developed by the equilibrium conditions of the generalized spatial truss model, by an analogy with Wagner’s model for steel beams of slender web (Onsongo, 1978). This model is again presented in Collins and Mitchell (1980) in order to emphasize the design for reinforced a prestressed concrete beam submitted to shear and the torsion. Vecchio and Collins (1986) added to the GDCFT the concrete tensile strength by studying the local stresses in the crack openings, and the softening of the concrete compressive strength associated with the concrete tensile strain. The theoretical model proposed in this paper does not consider the concrete tensile strength.

The author’s theoretical model includes the tensile strength of the CFRP in the cited models in order to increase the ultimate torsion. The CFRP maximum stress considered in the model is reduced, because the experimental results found in the literature show that the debonding of CFRP occurs before this material reaches the ultimate strength. The formulation of Chen and Teng (2003) is used to obtain the maximum value of the CFRP at the debonding.

Generalized diagonal compression field theory

This theory assumes that after the cracking, the concrete no longer has the capacity to resist to tension. The tensile forces are only resisted by the longitudinal and transverse reinforcements, and the compressive forces are resisted by the diagonal compression field with the strut angle θ, which is assumed equal to the crack angle.

The strength contribution of the CFRP reinforcements is added in the diagonal compression field model. These reinforcements only resist longitudinal and transverse tensions along with the steel reinforcements. Figure 1 shows all parameters of generalized spatial truss model for a reinforced concrete beam.

Figure 1.

Generalized Spatial Truss Model: (a) spatial truss model for reinforced concrete, (b) longitudinal and strut forces, (c) corner forces, (d) strut force, and (e) reinforced concrete beam submitted to torsion and strengthened with CFRP.

The equilibrium of the CFRP forces is performed in an analogous way to steel reinforcements (Figure 1(e)), but differing transversely by the spacing between the CFRP stirrups, $s_{ft}$ and longitudinally at the perimeter, $p_{c}$ . For reinforced concrete beams externally reinforced with CFRP these parcels must be added on generalized spatial truss model, then:

T = 2 A_{o} \sqrt{[\frac{Σ (A_{sl}) f_{sl}}{p_{o}} + \frac{Σ (A_{fl}) f_{fl}}{p_{c}}] \times [\frac{A_{st} f_{st}}{s_{st}} + \frac{A_{ft} f_{ft}}{s_{ft}}]}

(1)

where

$T$ – torsion;

$A_{o}$ – area enclosed by the centerline of the shear flow;

$p_{o}$ – perimeter limited by the center line of the shear flow;

$p_{c}$ – outside perimeter of the cross-section;

$A_{sl}, A_{fl}$ – cross-sectional area of the longitudinal steel and reinforcement of CFRP;

$A_{st}$ , $A_{ft}$ – cross-sectional area of the transverse steel and reinforcement of CFRP;

$f_{sl}$ , $f_{fl}$ – longitudinal steel and CFRP longitudinal stress;

$f_{st}, f_{ft}$ – transverse steel and CFRP stirrups stress.

Considering a unit concrete cracked panel with thickness $t_{d}$ , and strains in the longitudinal and transverse directions, the strain Mohr’s circle gives the maximum and minimum strains and their directions defined by the angle θ (Figure 2).

Figure 2.

Cracked panel with the longitudinal and transversal strains, principal strains and strain Morh’s circle.

The strain compatibility considers an effective bond between steel-concrete and CFRP- concrete, then

t g^{2} θ = \frac{ε_{l} + ε_{2}}{ε_{t} + ε_{2}}

(2)

γ_{lt} = 2 (ε_{l} + ε_{2}) cotg θ

(3)

ε_{l} + ε_{t} = ε_{1} - ε_{2}

(4)

Where

$θ$ – crack angle;

$γ_{lt}$ – distortion;

$ε_{l}, ε_{t}$ – longitudinal strain and transverse strain;

$ε_{1}, ε_{2}$ – tension principal strain and compression principal strain.

The compression strain is considered positive.

Vecchio and Collins (1986) proposed a new constitutive relationship, that is, considered the reduction of the compressive strength by the tensile strain increase that occurs perpendicularly to the compression strain (Figure 3(a)). This model considers the tension softening versus strain. In the study developed by Rahal and Collins (1996) the softening of the compression strain is also considered (Figure 3(b)).

Figure 3.

Concrete compression strength versus ratio between tensile and compression strains (Vecchio and Collins, 1986) and concrete constitutive relationship (Rahal and Collins, 1996).

Figure 3 shows the material softening that is not only due to the terms associated to the concrete compression strength. The parameter of tensile strain was included in order to reduce the limits proposed.

The equations for the concrete constitutive relationship are:

f_{2} = f_{2 \max} [\frac{2 ε_{2}}{S ε'_{c}} - {(\frac{ε_{2}}{S ε'_{c}})}^{2}]

(5)

f_{2 m a x} = \frac{f_{c}^{'}}{0.8 + 0.34 ε_{1} / ε_{c}^{'}}

(6)

S = \frac{f_{2 \max}}{f'_{c}}

(7)

where

$f'_{c}$ – concrete compressive strength (cylinder test);

$f_{2}, f_{2 \max}$ – concrete principal stress and maximum concrete stress;

$ε'_{c}$ – strain related to the stress $f'_{c}$ .

The elastic-plastic steel constitutive relationship is gives by (Figure 4(a)),

f_{s} = ε_{s} E_{s}

(8)

f_{s} = f_{sy}

(9)

where

$f_{s}$ – steel stress;

$ε_{s}$ – steel strain;

$E_{s}$ – modulus of elasticity of the steel;

$f_{sy}$ – steel yield stress.

Figure 4.

Steel (a) and CFRP (b) constitutive relationship.

The CFRP constitutive relationship is linear, and the ultimate strain is linked with ultimate stress $f_{fu}$ , or axial effective stress $f_{f, e}$ , which is coupled to the bond concrete-CFRP (Figure 4). The proposed theoretical model adopts the equation limited by the minimum value between the cited stresses, then:

f_{f} = ε_{f} E_{f} \leq f_{fu} or f_{f, e}

(10)

where

$f_{f}$ – CFRP axial stress;

$f_{f, e}$ – CFRP effective axial stress;

$f_{fu}$ – CFRP ultimate stress;

$ε_{f}$ – CFRP strain;

$E_{f}$ – modulus of elasticity of the CFRP.

The equation (9) is used to both directions of CFRP reinforcements, where the index $l$ is related to the longitudinal direction and index $t$ to the transverse one.

An epoxy resin, to guarantee the efficiency of the link, carries out the bond between the CFRP reinforcements and the concrete. Therefore, according to the tests results obtained in the literature, CFRP usually does not reach its maximum tensile strength, since the concrete substrate material detachment occurs. In Chen and Teng (2001) are described some fracture mechanics models complemented in Chen and Teng (2003) in order to determine the axial CFRP effective stress, which considers the debonding, then:

f_{f, e} = D_{f} σ_{f, \max} \leq f_{fu}

(11)

This model is based on the effective bond length, ultimate bond strength, and the stress distribution factor, respectively, given by:

L_{e} = \sqrt{\frac{E_{f} t_{f}}{\sqrt{f'_{c}}}}

(12)

σ_{f, \max} = 0.427 β_{w} β_{L} \sqrt{\frac{E_{f} \sqrt{f'_{c}}}{t_{f}}}

(13)

D_{f} = 1 - \frac{π - 2}{π λ}

(14)

where

$D_{f}$ – CFRP stress distribution coefficient;

$σ_{f, \max}$ – CFRP maximum axial stress;

$t_{f}$ – CFRP thickness;

$β_{L}, β_{w}$ – dimensionless coefficients;

$λ$ – normalized bond length coefficient.

Equation (13) is adopted for when the normalized bond length factor is $λ > 1$ , if $λ \leq 1$ the equation in Chen and Teng (2003) needs to be verified. The ratio between the maximum bond length, which in the case of the wrapped section is the height of the section, and with the effective bond length it follows:

λ = \frac{h}{L_{e}}

(15)

where

$h$ – beam height;

$L_{e}$ – effective length of the bond between concrete and CFRP.

For λ < 1, see Chen and Teng (2003). When $λ \geq 1$ consider the bond length coefficient $β_{L} = 1$ . The coefficient of reinforcement width $β_{w}$ with slope of 90° is given by

β_{w} = \sqrt{\frac{2 - w_{f t} / s_{f t}}{1 + w_{f t} / s_{f t}}}

(16)

where

$w_{ft}$ – CFRP stirrup width;

$s_{ft}$ – CFRP stirrup spacing.

This model analyzes the stirrups CFRP reinforcement or lateral bonded strips, so the reinforcements with orientation of 0° are not included. Therefore, in this paper the reduction of the maximum stress by the bond for reinforcement with orientation 0° is not considered in the theoretical model.

The wall thickness of the full section $t_{d}$ is defined to resist the compression stress generated by the shear flow q. Figure 5 shows the strain variation, the truly compression stress and the stress rectangular block of the concrete strut.

Figure 5.

Stress rectangular block of the concrete strut: (a) strain, (b) stress, and (c) retangular stress block.

Collins and Mitchell (1980) give expressions the following expression adopting the stress rectangular block of the concrete strut, so

β_{1} = \frac{4 - \frac{ε_{2}}{S ε'_{c}}}{6 - 2 \frac{ε_{2}}{S ε'_{c}}}

(17)

α_{1} = \frac{\frac{ε_{2}}{S ε'_{c}} - \frac{1}{3} {(\frac{ε_{2}}{S ε'_{c}})}^{2}}{β_{1}}

(18)

α_{1} f_{2 \max} = \frac{q}{a_{o} \sin θ \cos θ}

(19)

where

$β_{1}$ – coefficient for equivalent width of wall for the stress rectangular block;

$α_{1}$ – coefficient for the stress rectangular block.

The wall thickness is a function of the equivalent thickness $a_{o}$ , which is calculated by the equilibrium of the concrete, steel and CFRP strengths, thus

a_{o} = (\frac{Σ (A_{sl}) f_{sl}}{p_{o}} + \frac{Σ (A_{fl}) f_{fl}}{p_{c}} + \frac{A_{st} f_{st}}{s_{st}} + \frac{A_{ft} f_{ft}}{s_{ft}}) \frac{1}{α_{1} f_{2 \max}}

(20)

The equations related to the perimeter and the area bounded by the center line of the shear flow are, respectively:

A_{o} = A_{c} - \frac{1}{2} p_{c} a_{o} + a_{o}^{2}

(21)

p_{o} = p_{c} - 4 a_{o}

(22)

and the area of concrete and perimeter are given, respectively, by

A_{c} = bh

(23)

p_{c} = 2 (b + h)

(24)

where

$b$ – beam width;

$h$ – beam height.

With the expression for torsion angle gives by Onsongo (1978), and the expression of the equivalent thickness as a function of the torsion angle, follow

ψ = γ_{lt} \frac{p_{c}}{2 A_{c}}

(25)

a_{o} = β_{1} \frac{ϵ_{2}}{ψ \sin 2 θ}

(26)

where

$A_{c}$ – cross section area of the concrete;

$ψ$ – torsional angle per unit length;

$γ_{lt}$ – distortion of cross section;

$β_{1}$ – coefficient for equivalent width of wall for the stress rectangular block.

Collins and Mitchell (1980) report that the wall of the diagonal compression field should be calculated from center of the stirrups, because the covering resists until cracking occurs, then is neglected. However, this paper will use the entire concrete area, because other authors suggest this consideration, beside that the beam is reinforced externally with CFRP, which guarantees this integrity, then

a_{o} = \frac{1}{α_{1} f_{2 \max}} (\frac{Σ (A_{sl}) f_{sl}}{p_{o}} + \frac{Σ (A_{fl}) f_{fl}}{p_{c}}) \frac{\sin θ \cos θ}{\tan θ}

(27)

a_{o} = \frac{1}{α_{1} f_{2 \max}} (\frac{A_{st} f_{st}}{s_{st}} + \frac{A_{ft} f_{ft}}{s_{ft}}) \frac{\sin θ \cos θ}{\cot θ}

(28)

ε_{l} = [\frac{α_{1} β_{1} f_{2 \max} A_{c}}{2 p_{c} (\frac{Σ (A_{sl}) f_{sl}}{p_{o}} + \frac{Σ (A_{fl}) f_{fl}}{p_{c}})} - 1] ϵ_{2}

(29)

ε_{t} = [\frac{α_{1} β_{1} f_{2 \max} A_{c}}{2 p_{c} (\frac{A_{st} f_{st}}{s_{st}} + \frac{A_{ft} f_{ft}}{s_{ft}})} - 1] ϵ_{2}

(30)

The orthogonal strain to the direction of the strut diagonal compression field s given by Mohr’s circle, then:

ε_{1} = ε_{l} + ε_{t} + ε_{2}

(31)

The interactions of the model are calculated by flowchart, presented at Figure 6.

Figure 6.

Flowchart of interactions.

The elastic limit of the beam is defined by the torsion and torsion angle at cracking. These parameters allow to consider the complete beam behavior on the theoretical model. The torsional cracking equation adapted for the International System of Units is taken from

Rahal and Collins (1996) furnish for the torsional cracking equation the following equation (SI units):

T_{cr} = \frac{5}{12} \frac{A_{c}^{2}}{p_{c}} \sqrt{f_{c}^{'}}

(32)

For gives the torsion angle of cracking, it is necessary to estimate the tube wall thickness $t_{d}$ . The latter was estimated according to the first strain of the Diagonal Compression Field with Stress Softening (DCFSS)-CFRP model, so the cracking torsional angle and the tube wall thickness $t_{d}$ are get, respectively, by

ψ_{cr} = \frac{T_{cr} p_{o}}{4 {A_{o}}^{2} G t_{d}}

(33)

t_{d} = \frac{0, 50 A_{o}}{p_{o}}

(34)

where

$G$ – transverse elastic modulus;

$T_{cr}$ – torsional cracking;

$t_{d}$ – strut width.

The latter was estimated according to the first strain of the DCFSS-CFRP model. Before using the flowchart, it is necessary to calculate the parameters $T_{cr}$ and $ψ_{cr}$ . Following it is necessary to select the strut strain $ε_{2}$ and to vary from the null value until the maximum strain. Solving the equations described in this chapter from each strain selected the follow T and ψ for each $ε_{2}$ . The resolution needs two interactive processes, for the diagonal compression field thickness $a_{o}$ and for the strut tensile strain $ε_{1}$ .

The model verifies the CFRP debonding of concrete, and if this happens the process is completed.

Analysis of theoretical results

The theoretical was implemented in the software Mathcad v.15 and has as answer the graph of torsion versus torsional angle. The beams used in the implementation resemble section shape, shape and type of reinforcement. The experimental data and results analyzed were obtained from the beams tested by Ameli et al. (2004), Chalioris (2006a), Ghobarah et al. (2002), Hii and Al-Mahaidi (2006), Mohammadizadeh et al. (2009) and Silva Filho (2007).

Experimental results from the literature refer to externally CFRP reinforced beams with stirrups, which vary in width, spacing and number of layers. The sections of the beams were approximately twice as high as the width, noting that only the beam tested by Hii and Al-Mahaidi (2006) was width bigger than the height. The beams had stirrups, upper and lower longitudinal steel bars, and in some samples the beams had lateral longitudinal reinforcement.

Silva Filho (2007) shows two sets of three beams tested, in each one the beams contained the same concrete, reinforcement and configuration of reinforcement. The choice of beams to be analyzed was according to the proximity of the average results obtained from the set of the beams.

The all beams differ in dimensions, reinforcements, concrete strength and steel mechanical and geometrical parameters (Tables 1 and 2).

Table 1.

Geometric and mechanics parameters of analyzed beams.

Beams	Dimension			Concrete	Reinforcement
	b (cm)	h (cm)	Cover(cm)	f’_c (MPa)	ϕ_l (mm)	f_sly(MPa)	ϕ_t (mm)	s_st (cm)	f_sty (MPa)
VT1 (Silva Filho, 2007)	20	40	2	36.6	6 of 12.5	612.7	10.0	15.0	567.1
VTL3 (Silva Filho, 2007)	20	40	2	36.6	6 of 12.5	612.7	10.0	15.0	567.1
BCS1 (Mohammadizadehet al., 2009)	15	35	2.5	78.5	4 of 14.0	397.0	8.0	8.0	480.0
ACS1 (Mohammadizadehet al., 2009)	15	35	2.5	74.4	4 of 10.0	352.0	8.0	8.0	480.0
FS050D2 (Hii and Al-Mahaidi, 2006)	50	35	1.5	56.4	14 of 10.0	398.0	6.0	12.5	426.0
CFS (Ameli et al., 2004)	15	30	2.5	39.0	4 of 16.0	500.0	6.0	8.0	250.0
C2 (Ghobarah et al., 2002)	15	35	3.5 vert. 2.5 side	37.0	2 of 11.3, 2 of 16.0	409.0	6.3	7.0	456.8
RaS-FS150 (Chalioris, 2006a)	10	20	1.5	27.5	4 of 8.0	560.0	5.5	13.0	350.0
RbS-FS200 (Chalioris, 2006a)	15	30	1.5	27.5	4 of 8.0	560.0	5.5	10.0	350.0

Table 2.

CFRP parameters of analyzed beams.

Beams	Carbon fiber composites
	Width (L) (cm)	Thickness (e) (mm)	s_ft (cm)	Layers (n) (un.)	f_fu (MPa)
VT1 (Silva Filho, 2007)	15	0.122	30	2	3438
VTL3 (Silva Filho, 2007)*	15	0.122	30	1	3438
BCS1 (Mohammadizadeh et al., 2009)	10	0.176	20	1	3800
ACS1 (Mohammadizadeh et al., 2009)	10	0.176	20	1	3800
FS050D2 (Hii and Al-Mahaidi, 2006)	5	0.176	17.5	2	3800
CFS (Ameli et al., 2004)	10	0.165	20	1	3900
C2 (Ghobarah et al., 2002)	10	0.165	20	1	3550
RaS-FS150 (Chalioris, 2006a)	15	0.110	30	2	3900
RbS-FS200 (Chalioris, 2006a)	20	0.110	40	1	3900

Reinforced longitudinally with four 15 cm wide strips.

The results presented in Table 3 are experimental and analytical for the studies of ultimate torsion and the torsion angle, related to that torque. The values vary due to the different beams characteristics (Tables 1 and 2). The objective of choosing these beams is to show how the theoretical model (ultimate value) applies and behaves in different situations.

Table 3.

The ratio between the experimental results and theoretical model results.

Beams	Experimental result		Theoretical model		Ψ_EXP/Ψ_MOD	T_EXP/T_MOD
	Ψ_EXP (rad/m)	T_EXP (kN m)	Ψ_MOD (rad/m)	T_MOD (kN m)
VT1 (Silva Filho, 2007)	0.040	31.60	0.055	31.62	0.737	0.999
VTL3 (Silva Filho, 2007)	0.049	31.70	0.055	31.55	0.889	1.005
BCS1 (Mohammadizadehet al., 2009)	0.096	20.50	0.073	24.93	1.310	0.822
ACS1 (Mohammadizadehet al., 2009)	0.076	15.83	0.080	17.64	0.954	0.897
FS050D2 (Hii andAl-Mahaidi, 2006)	0.053	91.00	0.045	71.90	1.183	1.266
CFS (Ameli et al., 2004)	0.087	21.70	0.058	15.80	1.511	1.373
C2 (Ghobarah et al., 2002)	0.110	14.00	0.067	18.00	1.648	0.778
RaS-FS150 (Chalioris, 2006a)	0.100	4.33	0.100	3.46	1.002	1.251
RbS-FS200 (Chalioris, 2006a)	0.080	9.80	0.077	8.46	1.041	1.158
				Mean	1.14	1.06
				SD	0.30	0.21
				CV %	26.20	19.88

The proposed model, developed in Mathcad v.15, shows the behavior of the beams along the load history. It is identified the threshold at the time of the cracking that makes evident the limit of the section to consider the rectangular section as homogeneous, and, later, it is generated the stretch for which the theory of the field of diagonal compression for torsion is valid. The shear flow is resisted by the “wall” of the section, by the steel reinforcements and by the CFRP reinforcements.

Table 3 also shows the ratio between the experimental results and theoretical model results for torsional angle, torsion, and the mean, standard deviation and coefficient of variation values of these parameters.

The beams behavior is presented by the graphs $T x ψ$ in Figures 7 to 15, in which the graphs were generated with the experimental results and with the theoretical model. In order to make a comparison between the values, overlapping curves were plotted, making it possible to analyze the relevant shapes and points of these curves.

Figure 7.

Beam VT1 of Silva Filho (2007).

Figure 8.

Beam VTL3 of Silva Filho (2007).

Figure 9.

Beam BCS1 of Mohammadizadeh et al. (2009).

Figure 10.

Beam ACS1 of Mohammadizadeh et al. (2009).

Figure 11.

Beam FS050D2 of Hii and Al-Mahaidi (2006).

Figure 12.

Beam CFS of Ameli et al. (2004).

Figure 13.

Beam C2 of Ghobarah et al. (2002).

Figure 14.

Beam RaS-FS150 of Chalioris (2006a).

Figure 15.

Beam RbS-FS200 of Chalioris (2006a).

It is understood that in the comparison between the experimental results and those of the model, for the behavior there was inconsistency for the C2 beams (Ghobarah et al., 2002), ACS1 and BCS1 (Mohammadizadeh et al., 2009) in relation to the other beams under study. So, it was identified that these beams have the geometric ratios width and height b/h < 0.5 and the ratio between the cover and the smaller dimension of the d/cob element ≤6.

The study carried out shows the relevance of the cover of the reinforced concrete beam in the solution of the proposed model. The model developed in this work considers for the calculation of geometric parameters of the section all the covering of the beam.

Therefore, a new analysis was made consisting of two arguments: do not consider the half of the cover or the total cover. They are considered in the calculation for the perimeters and areas. Then:

p_{cc} = 2 (b + h - 4 cover)

(35)

A_{occ} = (b - a_{o} - 2 cob) \times (h - a_{o} - 2 cover)

(36)

p_{occ} = 2 (b + h - 2 a_{o} - 4 cover)

(37)

Substituting these parameters adapted by the corresponding parameters in the equations presented in item 3, we have the adjusted model for the consideration expressed in this new approach.

In order to complement the new model, we used the suggestion of Ameli and Ronagh (2007) for the CFRP strain, considering that the maximum strain is not on the outer face of the section, thus:

ε_{f} = ε_{s} (\frac{cover + t_{d}}{cover})

(38)

Following are the graphs of beam behavior from the experimental results and the theoretical model are show in Figures 16 to 21.

Figure 16.

Beam BCS1 of Mohammadizadeh et al. (2009) not considering half of covering.

Figure 17.

Beam BCS1 of Mohammadizadeh et al. (2009) not considering the total cover.

Figure 18.

Beam ACS1 of Mohammadizadeh et al. (2009) not considering half of covering.

Figure 19.

Beam ACS1 of Mohammadizadeh et al. (2009) not considering the total cover.

Figure 20.

Beam C2 of Ghobarah et al. (2002) not considering half of covering.

Figure 21.

Beam C2 of Ghobarah et al. (2002) not considering the total cover.

The results and the ratio between the experimental result and the theoretical model are shown in Tables 4 and 5:

Table 4.

Experimental result and analytical model not considering half of the beams coverings of BCS1 and ACS1 by Mohammadizadeh et al. (2009) and C2 by Ghobarah et al. (2002).

Beams	Experimental results		Analytic model		Ratios
			Half of covering
	Ψ_EXP (rad/m)	T_EXP (kN m)	Ψ_MOD (rad/m)	T_MOD (kN m)	Ψ_EXP/Ψ_MOD	T_EXP/T_MOD
BCS1 (Mohammadizadehet al., 2009)	0.096	20.50	0.080	19.57	1.204	1.048
ACS1 (Mohammadizadehet al., 2009)	0.076	15.83	0.087	14.26	0.874	1.110
C2 (Ghobarah et al., 2002)	0.110	14.00	0.076	12.37	1.462	1.132

Table 5.

Experimental result and analytical model not considering total of the beams coverings of BCS1 and ACS1 by Mohammadizadeh et al. (2009) and C2 by Ghobarah et al. (2002).

Beams	Experimental results		Analytic model		Ratios
			Half of covering
	Ψ_EXP (rad/m)	T_EXP (kN m)	Ψ_MOD (rad/m)	T_MOD (kN m)	Ψ_EXP/Ψ_MOD	T_EXP/T_MOD
BCS1 (Mohammadizadehet al., 2009)	0.096	20.50	0.092	14.12	1.042	1.452
ACS1 (Mohammadizadehet al., 2009)	0.076	15.83	0.098	10.70	0.776	1.479
C2 (Ghobarah et al., 2002)	0.110	14.00	0.088	7.57	1.253	1.849

Based on the beams with b/h ≥ 0.5 and d/cover > 6, it was found that the experimental and theoretical results are consistent, in addition to generating conservative values for the ultimate torsion (Table 6).

Table 6.

Experimental and theoretical model results for beams with b/h ≥ 0.5 e d/cover > 6.

Beams		Ψ_EXP/Ψ_MOD	T_EXP/T_MOD
VT1 (Silva Filho, 2007)		0.737	0.999
VTL3 (Silva Filho, 2007)*		0.889	1.005
FS050D2 (Hii andAl-Mahaidi, 2006)		1.183	1.266
CFS (Ameli et al., 2004)		1.511	1.373
RaS-FS150 (Chalioris, 2006a)		1.002	1.251
RbS-FS200 (Chalioris, 2006a)		1.041	1.158
	Mean	1.06	1.18
	SD	0.27	0.15
	CV%	25.15	12.82

Therefore, about the analysis that modifies the model in consideration of the thickness of the shear flow and disregards the half or fully the cover, it is concluded that the model initially should not consider, in some cases, all the contribution of the cover.

Table 7 shows the ratios of Table 6 and the ratios of Table 4 determining the statistical parameters for this set.

Table 7.

Experimental and model results.

Beams		Ψ_EXP/Ψ_MOD	T_EXP/T_MOD
VT1 (Silva Filho, 2007)		0.737	0.999
VTL3 (Silva Filho, 2007)		0.889	1.005
BCS1 (Mohammadizadehet al., 2009)		1.204	1.048
ACS1 (Mohammadizadehet al., 2009)		0.874	1.110
FS050D2 (Hii and Al-Mahaidi, 2006)		1.183	1.266
CFS (Ameli et al., 2004)		1.511	1.373
C2 (Ghobarah et al., 2002)		1.462	1.132
RaS-FS150 (Chalioris, 2006a)		1.002	1.251
RbS-FS200 (Chalioris, 2006a)		1.041	1.158
	Mean	1.10	1.15
	SD	0.26	0.13
	CV%	24.00	11.09

According to Chen and Teng (2003) method, for the bond between CFRP-concrete substrate, is conclusive for beams with high torsion strength, since the CFRP-resisted portion must be of great proportion, so that can reach the tension provided in the model and, therefore, the material debonding.

Concluding remarks

Most tests results behavior is similar and above the curves obtained with the theoretical model.

The beam C2 experimental curve of Ghobarah et al. (2002) is below the curve of the theoretical model. This beam has asymmetry in longitudinal reinforcements and coverings.

The torsional cracking, in general, is lower for the experimental results.

Beams with two layers had similar behavior in both curves.

Beams in which the width and height ratio is less than 0.5 have the experimental results for torsion below the result of the theoretical model.

It is understood that the thickness of the cover influences the efficiency of the theoretical model. A study was carried out in order to adapt this model, where it is considered the contribution of half or total of the cover. It was observed that the curves for both hypotheses lie below the experimental curves. A better accuracy was also found for not accounting for half the cover.

The parameters ratios in general are superior to unity, which is desirable because it is in favor of safety.

For ratios less than unity, the results are generally close to that value.

The results provided by the theoretical model have better accuracy for the torsion and is not as effective for the torsional angle.

The mean of all ratios is greater than unity.

The ultimate torsion has a coefficient of variation below 20%, which is considered one acceptable limit, however the coefficient of variation of the torsional angle has exceeded this limit.

The standard deviation of the ratios varies between 0.20 and 0.30, but the beams analyzed in this paper have different characteristics.

Figures represent behavior during loading, but the key purpose of the model is found the ultimate values, which is explicit in the all table data. The coefficients of variation obtained for the ultimate values are all less than 15% for torsion and 25% for torsion angle, which validates the consistency of the results obtained with the theoretical model, despite the appearance of the figures.

The results of the theoretical model for CFRP-reinforced beams applied to torsion when compared to the experimental results searched in literature, it is concluded that: the efficiency of the theoretical model depends on the geometric ratios width and height of the beam b/h, and of the ratio between the smallest dimension and the covering of the element d/cover.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Leandro Ferreira Silva

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