Stress–strain model for FRCM confined concrete elements

Composites: Part B 45 (2013) 1351–1359

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Stress–strain model for FRCM confined concrete elements Tomasz Trapko ⇑ Institute of Building Engineering, Wroclaw University of Technology, Pl. Grunwladzki 11, 50-377 Wroclaw, Poland

a r t i c l e

i n f o

Article history: Received 2 May 2012 Received in revised form 7 June 2012 Accepted 3 July 2012 Available online 13 July 2012 Keywords: A. Fibres A. Laminates B. Strength C. Analytical modelling

a b s t r a c t A proposition of a stress–strain model for concrete confined by fibre reinforced cementitious matrix (FRCM) and carbon fibre reinforced polymer (CFRP) composites has been presented. The model has been developed based on proprietary experimental study of cylindrical short elements confined with FRCM and CFRP jackets. The finished model includes influences of the composite strengthening, ultimate stress, critical strain and ultimate strain. The model assumes the stress–strain relationship has twofold characteristics – non-linear at first, when the cross-section carries the load and then linear, when the cross-section stiffens and full composite action occurs. The proposed stress–strain model has been experimentally verified. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction One of the most innovative and constantly developed methods of reinforcing structures is wrapping concrete with composite fibres. Different types of fibres and adhesives are used for different confinements. Widely known and popularly used are the fibre reinforced polymers (FRPs) built from materials woven using: carbon fibres CFRP, glass fibres GFRP and aramid fibres AFRP, bonded with the concrete using epoxy resin. Novelty is the p-phenylene benzobisoxazole PBO fibre, bonded together with a mineral matrix. It is used to create the fibre reinforced cementitious matrix (FRCM) confinement. Epoxy bonding agent in FRP composite makes it highly temperature-sensitive. Epoxy resins are safe to use to a limited extent in temperatures as high as +50 °C. Proprietary experiments showed that epoxy resin degrades already in about +30 °C. It then becomes impossible to reliably foresee the state of strain in compressed elements [1,2]. It is not the debonding temperature of the composite, but the temperature the resin plasticizes and ceases to serve its purpose i.e. of the adhesive bonding composite and concrete together. Further research concluded that compressive load capacity of specimens heated up to +60 °C is up to several dozen per cent lower compared to elements in +20 °C. Hence, the impact of temperature generated by production technologies used at given facility, fire as well as solar radiation on reliability of designed reinforcements has to be factored in. Concrete surfaces exposed to sunlight can reach temperatures as high as several dozen centigrade, depending on the colour of protective shell covering the composite. FRCM seems to address this problem. It uses ⇑ Tel.: +48 713203548; fax: +48 713221465. E-mail address: [email protected] 1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2012.07.001

mineral adhesive as mortar displaying properties similar to concrete. Hence the solution is fireproof and resistant to temperature exposure. The reinforced elements increase their load-bearing capacity due to composite confinement – the bond between the concrete and the jacket is established by using an adhesive. The increment depends on reinforcement intensity, defined as the number of strengthening composite layers. Transverse wrapping causes the concrete core to develop a complex state of stress translating directly to increase in load-bearing capacity. In case of composite confined cylinders, the lateral confining stress is homogeneous in all directions, as given by (Fig. 1):

fl ¼ r2 ¼ r3 ; r1 > r2 ¼ r3

ð1Þ

Mohr expanded on the Coulombe’s classic maximum normal stress theory by creating a triaxial state of stress through linking material failure not only with maximum shear stress but also the normal stress [3]. According to Mohr, state of stress during failure can be represented by the stress circles which share a common envelope – the failure envelope. By linking the effort level with shear and normal stress, Mohr obtained a curvilinear sn(r) relationship as opposed to Coulombe, who represented the sn–r relationship as rectilinear, and Tresca, who assumed that sn = const. (Fig. 2). Based on empirical data, stress circles can be constructed and then the Mohr failure envelope determined i.e. a best-fit line connecting the failure values of normal and shear stress for several Mohr circles. Laboratory experiments prove, there is no one, unequivocally best failure envelope, but there are multiple ones, allowing to generalise the theory [3]. According to Mohr, failure envelope can be accurately assumed as a straight line tangential

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Nomenclature Aa Acc

area of longitudinal steel gross area of the column section excluding area of longitudinal steel area of the effectively confined concrete core diameter of circular column initial tangent modulus of elasticity secant modulus at the point of maximum compressive stress fcc0 compressive strength of confined concrete compressive strength of unconfined concrete stress at critical strain of confined concrete ultimate strength of FRP/PBO clear spacing between transverse CFRP/PBO bands

Ae D Eo Esec fcc0 0 fco fk fu s0

to stress circles, where the slope of the tangent line corresponds to the concrete’s angle of friction /. Fig. 3 shows linear approximation of Mohr failure envelope, which Menne used to derive the relationship between principle stresses [4]. Based on Fig. 3 sin/: r1 r3

sin / ¼ r1 þ2 r3 2

 r20  r20

ð2Þ

By rearranging the Eq. (2) with respect to stress r1:

r1 ¼ r0 þ

1 þ sin / r3 ¼ r0 þ tg 2 ð45 þ 0:5/Þr3 1  sin /

r1 ¼ r0 þ k1 r3

ð3Þ ð4Þ

Compressive strength of confined concrete fcc0 (r1) is the sum of 0 compressive strength of unconfined concrete fco (r0), and the lateral confining stress fl (r2 = r3) multiplied by confinement coefficient k1. The fl stress depends of the shape of composite-wrapped core, tensile strength of composite and reinforcement intensity (number of composite sheet layers).

fcc0 ¼ fco0 þ k1 fl

ð5Þ

Fig. 4 shows a typical stress–strain curve for confined concrete (composite-wrapped) and unconfined concrete, plotted for circular specimens [5]. In order to determine contact surface lateral confining stress fl, it was assumed that maximum composite stress reaches its ultimate stress fu. This assumption holds only for circular cross-sections (6). Polygonal cross-sections, with rounded corners, require a different approach which would take account of stress concentration at composite kinks.

Fig. 1. Spatial state of stress in composite confined cylinder.

t k1 ke

total thickness of CFRP/PBO jacket concrete strength enhancement coefficient confinement effectiveness coefficient strain at maximum stress of confined concrete strain at maximum stress of unconfined concrete critical strain for confined concrete ultimate concrete compressive strain defined as strain at CFRP/PBO failure in tension fcu ratio of longitudinal reinforcement Aa to the area of cross-section Ac stress tensor

ecc eco ek ecu qcc r

fl ¼

2  t  fu ¼ 2  qf  fu D

ð6Þ

In case of multi-layer confinements, jacket thickness t is assumed to be the sum of constituent layer thicknesses. Local increases in composite thickness are neglected as caused by finishing capping. Confinement coefficient k1 is an empirical quantity, determined through laboratory experiments. In the literary reference, dozens of computational models can be found for modelling FRP confined compressed elements [6–18]. They all concentrate on describing analytically behaviour of particular elements and they use different confinement coefficients k1 = 2.0–4.0. Only cylindrical confinements, wrapped around the entire side surface of cylinder element are perfectly effective. In that case, as previously mentioned, lateral confining stress is homogeneous in all directions. Otherwise, to compensate for confinement ineffectiveness the confinement effectiveness coefficient ke have to be used.

fl0 ¼ ke fl

ð7Þ

The confinement effectiveness coefficient is defined as area of the effectively confined concrete core Ae to area of concrete core excluding the area of longitudinal reinforcement Acc.

ke ¼

Ae Acc

ð8Þ

For circular cross-section with ineffective transverse reinforcement areas i.e. reinforcement using composite strips instead of continuous jacket, the following is true [19]:

Fig. 2. Graphical representation of the maximum normal stress theory [3].

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is effectively reinforced. That effectiveness plunges down to 25% when transverse composite strips with spacing equal to diameter of cross-section are used. 2. Load-bearing capacity of concrete confined with composite sheet

Fig. 3. Spatial state of stress in composite confined cylinder [4].

Fig. 4. The r–e relationship for confined and unconfined concrete [5].

p  ðD  s Þ2 p  D2  ð1  s Þ2 s ð1  2D Þ2 Ae 2 2D ¼p 4 2 ¼ 4p 2 ¼ Acc 1  qcc  D  ð1  qcc Þ  D  ð1  qcc Þ 0

ke ¼

4

0

0

ð9Þ

4

Finally, let us assume:

0

ke ¼

s ð1  2D Þ 1  qcc

ð10Þ

Fig. 5 shows the ke coefficient relative to spacing (centre to centre) between composite strips to diameter of cross-section. Wrapped with continuous strengthening, the entire cross-section

Based on proprietary experimental study, including load-bear0 ing capacity of specimens applied with unconfined fco (r0) and confined fcc0 (r1) state of stress, and contact surface lateral confining stress r2 = r3 = fl, given by Tables 4–7, envelopes of Mohr’s circles were prepared. Investigated concrete cylinders had diameter of 113 mm and height of 350 mm, later reduced to 300 mm to adapt them to available CFRP sheets. That way there was no need to pleat the sheet to cover the entire specimen. Table 1 presents properties of p-phenylene benzobisoxazole (PBO) fibres of the FRCM jacket and CFRP fibres used during laboratory experiments. Data comes from technical approvals and own research [20–22]. Laboratory experiments took place in two stages. At the first stage, the influence of transverse reinforcement intensity was investigated, without temperature exposure (Table 2). At the second stage, elements of equal transverse reinforcement intensity were exposed to elevated temperatures +60 °C, +120 °C and +180 °C, and then tested to failure (Table 3). Fig. 6 shows both type confinements at the test stand. The concrete cylinders were capped with steel plates to maintain parallelism between the loaded ends of the cylinders and load axiality. Fig. 7 shows typical failure modes of CFRP and PBO confined elements. Failure envelopes were determined, and then the angles to longitudinal axes r. Figs. 8–11 graphically represent analyses for elements exposed and not exposed to elevated temperature. The / angle of inclination of Mohr failure envelope for 20M type elements (FRCM) and 20W type elements (CFRP) is approx. 26°, thus the confinement coefficient is k1  2.5 (Figs. 8 and 9). The / angle of inclination of Mohr failure envelope for 60M, 120M and 180M type elements (FRCM) and 60W, 120W and 180W type elements (CFRP) is approx. 26.5°, thus the confinement coefficient is k1  2.6 (Figs. 10 and 11). The confinement coefficient was assumed as k1 = 2.5 to determine theoretical compressive strength of confined concrete fcc0 . Table 1 Technical parameters of PBO and CFRP fibres [20–22].

*

No.

Type of fibre

Tension strength (MPa)

Modulus of elasticity (GPa)

Elongation at break (%)

Fibre density (g/cm3)

Fabric design thickness (mm)

1

PBO

270

2.15

1.56

0.0455

2

CFRP

5800 (5270*) 4300

238

1.80

1.76

0.1310

Own research.

Table 2 Research programme – elements not exposed to temperature.

Fig. 5. The ke coefficient for different ineffective transverse reinforcement areas s0 / D.

Number of layers

0

1

2

3

Unconfined elements

–

–

–

CFRP confined elements

20C_1 20C_2 –

FRCM confined elements

–

20W1_1 20W1_2 20M1_1 20M1_2

20W2_1 20W2_2 20M2_1 20M2_2

20W3_1 20W3_2 20M3_1 20M3_2

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Table 3 Research programme – elements exposed to temperature. Temperature

+60 °C

+120 °C

+180 °C

Unconfined elements

60C_1 60C_2 60W1_1 60W1_2 60M1_1 60M1_2

120C_1 120C_2 120W1_1 120W1_2 120M1_1 120M1_2

180C_1 180C_2 180W1_1 180W1_2 180M1_1 180M1_2

CFRP confined elements FRCM confined elements

Table 4 Data for determining Mohr failure envelopes for 20M elements. Specimen

ro (MPa)

r1 (MPa)

r2 = r3 (MPa)

20C_1 20C_2 20M1_1 20M1_2 20M2_1 20M2_2 20M3_1 20M3_2

23.23 22.00 – – – – – –

– – 32.48 32.66 42.48 42.96 58.07 55.80

– – 4.24 8.49 12.73

Table 5 Data for determining Mohr failure envelopes for 20W elements. Specimen

ro (MPa)

r1 (MPa)

r2 = r3 (MPa)

20C_1 20C_2 20W1_1 20W1_2 20W2_1 20W2_2 20W3_1 20W3_2

23.23 22.00 – – – – – –

– – 52.07 54.38 81.91 75.82 99.51 94.77

– – 9.97 19.94

Fig. 6. CFRP and FRCM confined concrete cylinders at the test stand.

29.91

Table 6 Data for determining Mohr failure envelopes for 60M, 120M and 120M elements. Specimen

ro (MPa)

r1 (MPa)

r2 = r3 (MPa)

60C_1 60C_2 120C_1 120C_2 180C_1 180C_2 60M1_1 60M1_2 120M2_1 120M2_2 180M3_1 180M3_2

25.19 24.48 24.37 24.87 20.89 22.65 – – – – – –

– – – – – – 37.32 36.45 33.36 33.13 34.71 34.34

– – – – – – 4.24 4.24 4.24

Table 7 Data for determining Mohr failure envelopes for 60W, 120W and 120W elements. Specimen

ro (MPa)

r1 (MPa)

r2 = r3 (MPa)

60C_1 60C_2 120C_1 120C_2 180C_1 180C_2 60W1_1 60W1_2 120W2_1 120W2_2 180W3_1 180W3_2

25.19 24.48 24.37 24.87 20.89 22.65 – – – – – –

– – – – – – 53.60 54.79 52.87 48.19 58.72 53.22

– – – – – – 9.97 9.97 9.97

Fig. 7. Typical failure of CFRP and FRCM confined concrete cylinders in a compression test.

T. Trapko / Composites: Part B 45 (2013) 1351–1359

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Fig. 8. Mohr failure envelope for 20M elements.

Fig. 9. Mohr failure envelope for 20W elements.

Figs. 12 and 13 show the relationship between lateral stress fl and compressive strength of confined concrete fcc0 relative to compressive strength of unconfined concrete fc0 . Chart 12 shows failure

envelope for elements with temperature exposure confined with PBO mesh (FRCM) – 20M_e and 20M_t data and for elements

Fig. 10. Mohr failure envelope for 60M, 120M and 120M elements.

Fig. 11. Mohr failure envelope for 60W, 120W and 120W elements.

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Fig. 14. Empirical and analytical stress for FRCM and CFRP confined elements.

Fig. 12. Compressive strength of confined concrete fcc0 relative to fl stress for elements not exposed to elevated temperature.

Fig. 15. Compressive strength of confined concrete fcc0 relative to fl stress for elements not exposed to elevated temperature.

Fig. 13. Compressive strength of confined concrete fcc0 relative to fl stress for elements exposed to elevated temperature.

Table 8 Empirical and analytical stress for FRCM and CFRP confined elements. Specimen

20M1_1 20M1_2 20M2_1 20M2_2 20M3_1 20M3_2 60M1_1 60M1_2 120M1_1 120M1_2 180M1_1 180M1_2 20W1_1 20W1_2 20W2_1 20W2_2 20W3_1 20W3_2 60W1_1 60W1_2 120W1_1 120W1_2 180W1_1 180W1_2

Empirical values

Analytical values

fcc0 (MPa)

fcc0 /fco

fcc0 (MPa)

fcc0 /fco

32.48 32.66 42.48 42.96 58.07 55.80 37.32 36.45 33.36 33.13 34.71 34.34 52.07 54.38 81.91 75.82 99.51 94.77 53.60 54.79 52.87 48.19 58.72 53.22

1.44 1.44 1.88 1.90 2.57 2.47 1.65 1.61 1.48 1.47 1.54 1.52 2.30 2.41 3.62 3.35 4.40 4.19 2.37 2.42 2.34 2.13 2.60 2.35

33.22 33.22 43.83 43.83 54.44 54.44 33.22 33.22 33.22 33.22 33.22 33.22 47.53 47.53 72.46 72.46 97.38 97.38 47.53 47.53 47.53 47.53 47.53 47.53

1.47 1.47 1.94 1.94 2.41 2.41 1.47 1.47 1.47 1.47 1.47 1.47 2.10 2.10 3.20 3.20 4.31 4.31 2.10 2.10 2.10 2.10 2.10 2.10

confined with CFRP sheet – 20W_e and 20W_t data. Coefficient of determination R2 = 0.97. Chart 13 shows failure envelope for elements exposed to elevated temperature confined with PBO mesh (FRCM) – M_e and M_t data and for elements confined with CFRP sheet – W_e and W_t data (e – empirical data, t – theoretical data). Coefficient of determination R2 = 0.91. Table 8 and Fig. 14 shows comparison between compressive strength of confined concrete fcc0 determined analytically using the formula (5) and its empirical values. Coefficient of determination, i.e. accuracy of the model – R2 = 0.96. The scaled confinement

Fig. 16. Compressive strength of confined concrete fcc0 relative to fl stress for elements exposed to elevated temperature.

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coefficient was taken as k1 = 2.5. Consequently the comparison between laboratory experiments and theoretical analysis gave satisfactory results. Obtained accuracy ranged between 0.8% and 12% (except for the 180W1_1 specimen, with 20.7%). Note that positive model accuracy was achieved both for PBO (FRCM) and CFRP confined specimens. 3. Strain in concrete confined with composite sheet The quantity fcc0 corresponds with strain at maximum stress of confined concrete ecc. Reference literature describes it using function depending on strain at maximum stress of unconfined concrete eco.



ecc ¼ eco  1 þ k2 

fl fco0

 ð11Þ

Fig. 15 shows scaling procedure of k2 coefficient for elements not exposed to elevated temperature confined with PBO mesh (FRCM) – 20M_e and 20M_t data and for elements confined with CFRP sheet – 20W_e and 20W_t data (e – empirical data, t – theoretical data). The k2 coefficient for FRCM confined elements was taken k2 = 10.5, whereas for CFRP confined elements k2 = 6.0. Coefficient of determination R2 = 0.95. Forecasting confined concrete strain for initially heat treated elements is a different ball game altogether. As Fig. 17 shows, load-bearing capacity of PBO confined elements is reached once critical strain ek = 3.5‰ is exceeded, and is temperature-invariant. Hence the k2 coefficient was scaled within 0 6 ecc 6 ek interval – k2 = 3.0 was assumed. Because strain measurements for a small sample were too scattered (9.162‰ to 13.710‰) in case of CFRP confined elements, that sample was deemed non-representative, and the k2 coefficient could not be unambiguously determined (Fig. 16). Table 9 and Fig. 18 compare the ecc strains measured during tests with values calculated from the relationship (11). Strain has been associated with corresponding control specimens. 20M and 20W – average for specimen group 20C, 60M and 60W – average for specimen group 60C, 120M and 120W – average for specimen group 120C, 180M and 180W – average for specimen group 180C. 4. The r–e relationship for elements confined with PBO mesh (FRCM)

Table 9 Empirical and analytical strain for FRCM and CFRP confined elements. Specimen

20M1_1 20M1_2 20M2_1 20M2_2 20M3_1 20M3_2 60M1_1 60M1_2 120M1_1 120M1_2 180M1_1 180M1_2 20W1_1 20W1_2 20W2_1 20W2_2 20W3_1 20W3_2 60W1_1 60W1_2 120W1_1 120W1_2 180W1_1 180W1_2

Empirical values

Analytical values

ecc (‰)

ecc/eco

ecc (‰)

ecc/eco

6.184 6.983 12.119 11.369 18.108 17.048 3.573 3.593 3.537 3.473 3.575 3.518 9.082 10.116 13.694 14.745 12.112 11.146 10.844 11.397 13.710 12.412 9.162 10.352

2.51 2.84 4.92 4.62 7.36 6.93 1.55 1.56 1.60 1.57 1.39 1.37 3.69 4.11 5.56 5.99 4.92 4.53 4.71 4.95 6.20 5.61 3.57 4.04

7.31 7.31 12.16 12.16 17.01 17.01 3.48 3.48 3.49 3.49 3.65 3.65 8.97 8.97 15.48 15.48 – – – – – – – –

2.97 2.97 4.94 4.94 6.91 6.91 1.51 1.51 1.52 1.52 1.58 1.58 3.65 3.65 6.29 6.29 – – – – – – – –

corresponding fk stress. Once the critical strain ek is exceeded the cross-section stiffens and full composite action occurs of external PBO confinement – the B–C phase. In that phase the r–e curve is linear, as Fig. 19 shows. Based on reviewed results of laboratory experiments it was assumed, that the r–e curve changes its course once the critical strain ek = 3.50‰ is reached. In order to define longitudinal stress in the concrete fc for elements subject to constant strain increment and load increment (quasi-static) during the A–B phase, the relationship given by Popovics was used [23]. During the B–C phase, the r–e curve depends on secant modulus of elasticity Esec,2.

fc ¼

fco0  x  r ! 0 6 ecc 6 ek r  1 þ xr

fc ¼ fco0 þ Esec;2  ðec  eco Þ ! ecc > ek

ð12Þ ð13Þ

where PBO (FRCM) confined cylinders exhibit twofold behaviour. In the A–B phase (Fig. 19), until the cross-section reaches critical strain ek, the specimen behaves as unconfined element reaching

Fig. 17. Experimental r–e relationship for elements 60M1, 120M1 and 180M1.

x¼

ec eco

ð14Þ

r¼

Eo Eo  Esec;1

ð15Þ

Fig. 18. Empirical and analytical strain for FRCM and CFRP confined elements.

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Fig. 19. The r–e relationship for elements confined with PBO mesh.

Fig. 21. The r–e relationship for PBO confined elements not exposed to temperature.

Fig. 20. The a coefficient scaling relative to (16).

Eo ¼ a  ðfco0 Þ Esec;1 ¼

Esec;2 ¼

1=3

fk

ek fcc0  fco0

ecc  eco

ð16Þ ð17Þ

ð18Þ

The modulus of elasticity Eo, featured in the model, depends on ini0 tial compressive strength of unconfined concrete fco . The value of the modulus was assumed to be the product of empirical coefficient a and compressive strength of unconfined concrete to the power of 1/3. Fig. 20 shows scaling procedure of the a coefficient based on empirical data. The lowest standard deviation between experimentally determined modulus of elasticity and analytically determined from (16) was the criterion for selecting the coefficient. Finally, let us assume:

Eo ¼ 2700  ðfco0 Þ1=3

ð19Þ

Given by (12) and (13) relationships hold, provided the lateral stress is homogeneous across the entire side surface of the concrete core. Otherwise, to compensate for confinement ineffectiveness the confinement effectiveness coefficient ke have to be used. Fig. 21 compares the r–e curves for selected test specimens not exposed to temperature (20M1_e, 20M2_e; 20M3_e – solid lines) with theoretical curves determined from the relationship (12) and (13) (20M1_t, 20M2_t; 20M3_t – dotted lines). Fig. 22 in turn, shows empirical and theoretical r–e relationships for specimens with temperature exposure (60M1_e, 120M2_e; 180M3_e – experimentally

Fig. 22. The r–e relationship for PBO confined elements exposed to temperature.

determined solid lines 60M1_t, 120M2_t; 180M3_t – analytically determined dotted lines). As with forecasting theoretical strain, strain curves were determined within 0 6 ecc 6 ek, which if exceeded the load-bearing capacity is reached. The proposed r–e computational model for concrete reinforced with mineral based composites is accurate and consistent with experimental results. Neither significant underestimation not overestimation of analytical results was observed compared to findings of laboratory experiments. This is also true for the Wang model [18]. Completed theoretical analyses [24] prove, the Wang model overstates the compressive strength of confined concrete fcc0 , and analytical results vary with empirical by 9–93%. This is caused by overestimated Poisson’s ratio m = 0.50 and overlooking that transverse strain at failure is usually lower than during tensile test of FRP composites. There are no papers concerning stress–strain model for FRCM confined concrete elements. There are, however, numerous models describing the r–e relationship for FRP confined concrete elements [6–18,25]. The paper [25] compares several calculation methods concerning cylinder specimens. Their absolute fcc0 error averaged 30%, whereas ecc error exceeds 100%. Many those models use

T. Trapko / Composites: Part B 45 (2013) 1351–1359

empirical coefficients aligned with research results. The proposed model attempts to avoid variable parameters and to create a more general relationship. It factors in the ineffective transverse reinforcement area, which has been missing from previous models. That model for CFRP confined concrete elements, given in [19], also includes longitudinal CFRP straps and different cross-sections of compressed elements. The theoretical model is accurate to a satisfactory extent, both in terms of calculating compressive strength (load-bearing capacity) of compressive strength of confined concrete fcc0 and forecasting the state of strain and its characteristics. 5. Conclusions This paper has proposed a computational model for FRCM confined concrete elements. The concrete confinement coefficient k1 has been determined for the classic relationship between principle stresses. The proposed model has been verified by laboratory experiments. The results of this study indicated that this model can effectively predict the behaviour of circular elements. Based on conducted studies, the following conclusions were drawn: 1.

2.

3.

4.

The initial heat treatment has no effect on the / angle of inclination of Mohr failure envelope. Hence the constant confinement coefficient for confined concrete could be determined. Based on the relationship (11) it is possible to forecast longitudinal strain for FRCM and CFRP confined elements not exposed to temperature. The relationship (11) is inaccurate for CFRP confined elements subject to initial heat treatment, as proven by [1,2]. To define the stress–strain curve for concrete confined by FRCM composites, the classic Popovics relationship has been used – adapted for bilinear characteristics stress–strain curve traces. The stress–strain curve in bilinear. During the first phase it is non-linear as in case of unconfined concrete until critical strain ek = 3.50‰. Then the curve becomes linear until failure.

References [1] Trapko T. The influence of temperature on the durability and effectiveness of strengthening of concrete with CFRP composites. Eng Build 2010;10:561–4 [in Polish].

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