Bending model for composite UHPFRC-RC elements including tension stiffening and crack width

Engineering Structures xxx (xxxx) xxxx

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Bending model for composite UHPFRC-RC elements including tension stiffening and crack width ⁎

Carlos Zanuy , Gonzalo S.D. Ulzurrun Department of Continuum Mechanics and Structures, E.T.S. Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Madrid (UPM), Spain

A R T I C LE I N FO

A B S T R A C T

Keywords: UHPFRC Composite structures Bending Strengthening Tension stiffening Cracking

The application of a thin ultra-high performance fiber-reinforced concrete (UHPFRC) layer at the tensile side of reinforced concrete (RC) elements has lead to an efficient structural concept with promising capacities. Composite UHPFRC-RC elements have an increased bending capacity with respect to the RC member. In addition, the serviceability and durability are also improved as the composite element has higher stiffness and smaller crack widths. In order to widen the application of composite UHPFRC-RC elements, practitioners and designers need analytical and numerical tools with a sound mechanical background, if possible conceptually analogous to well-established models for conventional concrete structures. The paper starts with a discussion of the main drawbacks of existing models for UHPFRC-RC elements in bending. Secondly, new experimental basis is provided with own tests where the digital image correlation (DIC) technique is exploited to understand the crack pattern development and interfacial behaviour of UHPFRC-RC. Finally, the main goal of the paper is the proposal of a new model for UHPFRC-RC elements in bending which includes tension stiffening and the interaction at the steel-concrete and concrete-UHPFRC interfaces by means of a new composite tension chord model. The tension chord model is based on the stress transfer mechanisms between adjacent cracks, which include the explicit consideration of bond stresses between the constituent materials (i.e. steel and concrete, and UHPFRC and concrete). The compatibility condition between the UHPFRC and the RC layers is established in terms of the extension of each layer between adjacent cracks, as the classic compatibility of strains based on perfect bond assumption cannot be used upon macrocrack formation. Rather, the extension of the UHPFRC layer includes the contribution of cross-sections within the elastic, hardening or softening stages of the UHPFRC. The proposed model allows estimating the whole bending response, including calculation of crack widths and average curvatures between cracks.

1. Introduction

(a) development of UHPFRC with moderate fiber contents of around 2% in order to reduce material price [3]; (b) local applications of UHPFRC in combination with conventional materials, for example, as strengthening layer of reinforced concrete (RC) [4]. The design of composite members combining UHPFRC and RC (hereafter, referred to as UHPFRC-RC) has become an efficient solution for retrofitting existing bridge decks. The application of a thin UHPFRC layer to the tensile side of the RC member has several advantages [5,6], as the amount of required UHPFRC is moderate, the excellent tensile properties of UHPFRC can be utilized, the low permeability of UHPFRC increases durability and the UHPFRC also contributes to shear strength. Because of its more recent development than conventional RC structures, the availability of mechanical models for UHPFRC-RC structures is less numerous. Design recommendations have been proposed [7,8], but some of them still rely on empirical results without a

Ultra-high performance fiber-reinforced concrete (UHPFRC) is a cement-based composite material containing a high content of steel fibers which has superior properties in compression and tension [1,2]. The compressive strength is typically of the order of 150 MPa for fiber volumetric contents of 4–6%. In tension, UHPFRC is able to develop a hardening stage characterized by uniform matrix microcracking before macrocracking localization. After macrocracking, the UHPFRC is still able to carry tensile stresses due to the bridging ability of the fibers. Despite its interesting material properties, some factors (e.g. high price, lack of specific design codes, and slow implementation of innovations in the construction sector) have limited the general structural application of UHPFRC. In order to take advantage of the characteristics of UHPFRC, researchers and engineers have focused on two working lines:

⁎

Corresponding author at: ETS Ingenieros de Caminos, c/ Profesor Aranguren, 3, 28040 Madrid, Spain. E-mail address: [email protected] (C. Zanuy).

https://doi.org/10.1016/j.engstruct.2019.109958 Received 16 July 2019; Received in revised form 18 October 2019; Accepted 18 November 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Carlos Zanuy and Gonzalo S.D. Ulzurrun, Engineering Structures, https://doi.org/10.1016/j.engstruct.2019.109958

Engineering Structures xxx (xxxx) xxxx

C. Zanuy and G.S.D. Ulzurrun

Notation

x neutral axis depth, abscissa y depth ordinate from top of the section ΔLc, ΔLs, ΔLU elongation of concrete, steel and UHPFRC layers Φ diameter of steel bars εc, εs, εU strain at concrete, steel and UHPFRC εcc, εpc elastic and hardening strain capacity of UHPFRC εc,cr, εs,cr, εU,cr strain at cracked section of concrete, steel and UHPFRC εcm, εsm, εUm average strain of concrete, steel and UHPFRC εc,mid, εs,mid, εU,mid stress at mid-section of concrete, steel and UHPFRC εy yield strain of steel κ curvature ρ steel reinforcing ratio ρU ratio between UHPFRC and concrete areas in tension σcc matrix cracking strength of UHPFRC σpc hardening strength of UHPFRC σc, σs, σU stress of concrete, steel and UHPFRC σc,cr, σs,cr, σU,cr stress at cracked section of concrete, steel and UHPFRC σc,mid, σs,mid, σU,mid stress at mid-section section of concrete, steel and UHPFRC τb bond stress between concrete and steel τU bond stress between concrete and UHPFRC

As area of steel reinforcement Ec, Es, EcU elastic modulus of concrete, steel and UHPFRC Ep, Epc hardening modulus of steel and UHPFRC F load GF fracture energy of UHPFRC LR reference length M bending moment N axial force b section width d section effective depth h section height fc, fcU compressive strength of concrete and UHPFRC fct tensile strength of concrete fy yield strength of steel hc, hU height of concrete and UHPFRC layers hc,eff effective depth of concrete in tension lcc, lpc elastic and hardening length of UHPFRC ls, ly elastic and yield length of steel sr distance between adjacent cracks u, v horizontal and vertical displacements w crack width wU crack width at the UHPFRC w0, wc crack width properties of UHPFRC in softening

mechanism is taken into account by considering bond stresses between the steel reinforcement and the conventional concrete, and also between UHPFRC and concrete. The model makes use of a condition of compatibility of extension between adjacent cracks, rather than requiring a reference length concept. The model can be used to calculate curvatures (both at the macrocrack section and the average curvature between adjacent cracks) and crack widths at the RC and UHPFRC layers. The model capabilities are compared with own and existing experimental results. Regarding own experiments, digital image correlation (DIC) technique is used to provide new insights which allow understanding the crack development, the interaction between UHPFRC and RC layers and measurement of crack widths.

mechanically consistent background. In the authors’ opinion, the lack of engineering tools to model UHPFRC-RC elements can be a disadvantage for a broader generalization of its structural use. In the present paper, the attention is paid to the bending response of UHPFRC-RC elements. Even though sectional models already exist [4,5], they are based on the concept of the reference length in order to consider the compatibility between the UHPFRC and RC components of the system. The reference length is a fictitious length to convert crack widths of the UHPFRC to strains, so that the compatibility condition that plain sections remain plane can be easily extended to the UHPFRC layer. A unique formulation with mechanical background to determine the reference length has not been reported so far. Moreover, empirical studies have shown that different values are to be used for calculation of curvatures (and, therefore, deflections) or crack widths [9]. The model proposed in the present paper considers the interaction between the UHPFRC and RC components with a stress transfer mechanism. The tension stiffening

(a)

hc d

h

(y)

y

M

c

(y)

x

As s

s

hU

(y)

U

(y)

b

c

s

fc

fy

(b)

wU

U

1

pc

Ep

cc

1

Es

Ec cp

GF

Ecu

1

1

pc

E pc

0

1

c

s

cc

pc U

Fig. 1. (a) Stress-strain state of a UHPFRC-RC section in bending; (b) Material models. 2

w0

wc wU

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where the intermediate point (w0, σ0) can be calculated to fit the area below the σU-wU curve to the fracture energy GF. Typically, w0 ≈ wc/5, and the crack width limit wc is ¼-½ of the fiber length. The two unknowns of the problem (εc,sup, x) can be calculated from the two equilibrium equations, as follows:

2. Problem statement. Bending response of UHPFRC-RC elements The bending behaviour of a UHPFRC-RC section can be understood with the bending moment-curvature diagram. In the present paper, the analysis is focused on a rectangular section with a thin UHPFRC layer at the tensile side. Longitudinal steel reinforcement is also embedded into the conventional concrete. Moreover, it is considered that the UHPFRC does not contain additional longitudinal reinforcing bars. The bending moment-curvature diagram can be solved by equilibrium, constitutive and compatibility equations. The strain and stress distribution of a UHPFRC-RC cross-section under an external bending moment M can be represented as plotted in Fig. 1(a). The strain distribution can be defined by two parameters, e.g. the strain at the top (εc,sup) and the neutral axis depth (x). Thus, the curvature (κ) and the strain at any fiber (ε(y)) are as follows:

κ=

y=h

N=0=

(2)

σcc EcU

εU = εpc +

(4)

where (εcc, σcc) represent matrix microcracking and (εpc, σpc) is the localization of a macrocrack. When a macrocrack opens, the UHPFRC enters a softening stage and its deformed state is characterized by a crack width (wU). A bilinear model is adopted for the softening stage:

σU = σ0 +

σU =

w 0 − wU (σpc − σ0), wU ⩽ w0 w0

(5)

wc − wU σ0, w0 < wU ⩽ wc wc − w0

(6)

2

250

wU LR

(9)

so that the compatibility condition Eq. (2) can be used for the UHPFRC layer after macrocracking localization under the assumption of perfect bond. As an example to understand the stages of UHPFRC-RC under bending, the behaviour of the cross-sections depicted in Fig. 2 is analyzed with the explained model (Eqs. (1)–(9) and Fig. 1). The dimensions and material properties are chosen without loss of generality, though they agree with the experiments analyzed in Section 3. A reference RC section without UHPFRC is considered, as well as two strengthened sections with 35 and 55 mm thick UHPFRC layers, respectively. All three cross-sections have the same total height. The bending moment-curvature and bending moment-crack width diagrams are represented in Fig. 3(a). Note that the diagrams corresponding to hU = 0 mm (section without UHPFRC) are calculated under the cracked section assumption. The crack width at the level of the steel reinforcement is not a direct result of the model and it is calculated in a simplified way in this Section, as follows:

(3)

σU = σcc + Epc (εU − εcc ), εcc < εU ⩽ εpc

(8)

where M is the applied bending moment and i = c, s, U depending of the material placed at depth y. The problem can be numerically solved by dividing the section into layers of constant thickness, whereby the integrals of Eqs. (7) and (8) are converted into sums. The main difficulty of the problem above arises when the UHPFRC enters in the softening stage, as its state is characterized by crack width rather than by strain. This problem was addressed by Habel [9] by proposing the concept of the reference length, LR. The reference length is used to convert the crack width into a strain as follows:

The stress distribution can be then obtained from the constitutive behaviour of each material (Fig. 1(b)). In the present paper, the nonlinear stress-strain equation of the Model Code [10] is adopted for the concrete in compression. In addition, it is assumed that the concrete cannot carry tensile stresses, i.e. the analysis is carried out at the cracked section. For the steel, a bilinear behaviour is considered. For the UHPFRC in tension, an elastic-hardening behaviour is considered in the first stage before macrocrack localization, when the deformation state can still be defined by strains (εU):

σU = EcU εU , εU ⩽ εcc =

∫ σi (y)(x − y) dAi y=0

(1)

ε (y ) = κ (x − y )

(7)

y=h

M=

εc,sup x

∫ σi (y) dAi y=0

(10)

w = εsm sr

2

16 250

16

2 250

58

55

35 125

hU = 0 mm

125

125

hU = 35 mm

hU = 55 mm

Concrete: fc = 41 MPa, Ec = 34.4 GPa,

cp

= 0.002

Steel: fy = 500 MPa, Es = 200 GPa, Ep = 5 GPa UHPFRC:

cc

= 4.6 MPa,

pc

= 6.2 MPa,

pc

= 0.0022, EcU = 53.4 GPa,

wc = 6 mm, GF = 9.9 kN/m Fig. 2. Cross-section and material properties of example. Dimensions in mm. 3

16

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45 35

2

30

5

3

25

20 15

hu = 0 mm hu = 35 mm hu = 55 mm

1

10

2

35 30

3

25

20 15

hu = 0 mm hu = 35 mm hu = 55 mm

1

10

5

5

0

0 0.00

0.02

0.04

0.06

0.00

0.02

Curvature, [m -1 ]

30

3

25

20

hu = 0 mm

15

hu = 35 mm

1

hu = 55 mm

5

2

35

5

30

3

25

20

hu = 0 mm

15

hu = 35 mm

1

10

hu = 55 mm

5

0

0 0.0

0.2 0.4 0.6 Crack width at steel, w [mm]

0.8

0.0

45

0.2 0.4 0.6 Crack width at steel, w [mm]

45

40 35

3

30 25

2

20 15

hu = 35 mm hu = 55 mm

10 5

0.8

3

40

4

Moment, M [m kN]

Moment, M [m kN]

0.06

4

40

2

Moment, M [m kN]

Moment, M [m kN]

45

4

40 35

0.04

Curvature, [m -1 ]

45

10

5

4

40

Moment, M [m kN]

40

Moment, M [m kN]

45

4

35 30 25

20

2

15

hu = 35 mm hu = 55 mm

10 5

0 0.0

0

0.2 0.4 0.6 Crack width at UHPFRC, wU [mm]

0.0

(a)

0.2 0.4 0.6 Crack width at UHPFRC, wU [mm]

(b)

Fig. 3. Bending moment-curvature and bending-moment crack width diagrams: (a) LR = 167 mm; (b) LR = 800 mm.

where εsm is the average steel strain between adjacent cracks and sr is the crack spacing. To calculate εsm, the formulation of the Model Code [10] has been used, and sr = 150 mm has been taken according to experimental evidence (refer to Section 3). According to Fig. 3(a), the contribution of the UHPFRC is evident, as the strengthening layer provides: smaller pre-peak deformations, larger peak bending moment and higher post-peak capacity. The stages of the UHPFRC-RC sections can be analyzed with the help of key points represented in the diagrams: Point ①: matrix microcracking. Strain εcc is reached at the UHPFRC. The strengthening layer enters the strain hardening stage. Point ②: UHPFRC macrocracking. Strain εpc is reached at the UHPFRC. The softening stage of UHPFRC begins. For higher bending moments the UHPFRC is cracked and Eq. (9) is used to transform crack widths to strains. Point ③: yielding of longitudinal steel bars. Point ④: the crack width of the UHPFRC reaches w0. Point ⑤: the concrete in compression reaches a strain of 3.5‰.

It should be noted that the former order of key points can vary depending on the cross-section configuration and the material properties of the UHPFRC. The diagrams of Fig. 3(a) have been obtained for a reference length of LR = 167 mm, according to the recommendation by AFGC [11] that LR = 2/3h. Habel [9] noted that very different values of LR were necessary either to determine curvatures or to calculate crack widths. Values of LR ≈ 10 h were necessary in some cases. To understand the important influence of LR, the analysis is repeated in Fig. 3(b) with LR = 800 mm. The differences between both calculations are evident from point ②, upon localization of macrocrack. The higher reference length produces a smaller contribution of the UHPFRC and larger crack widths. The results depend significantly on the value of LR. According to the discussion above, a model for UHPFRC-RC elements in bending is necessary to overcome the drawbacks of the reference length concept. It is convenient to introduce a mechanicallybased model so that both curvatures and crack widths can be calculated under the same assumptions. In the present paper, a tension chord model is proposed to take into account the interaction between 4

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UHPFRC and RC in the tension side of beams, with an appropriate compatibility condition. Previously, experimental results are presented in Section 3 to support the model assumptions.

Table 1 Material properties. Concrete Cement: CEM I 42,5R

3. Experimental approach

Cement content: 350 kg/m3 Abrams cone: 13–14 cm Cylinder compressive strength: 40.8 (testing age)

Max. Aggregate size: 12 mm

3.1. Overview of tests

Cylinder compressive strength: 38.1 MPa (28 days)

The bending behaviour of UHPFRC-RC elements is governed by crack formation and development. Since cracks can form at both the RC and UHPFRC layers, it is necessary to understand the interlayer interaction. The attention is paid at bending behaviour, which means that additional effects produced by shear-bending interaction are not dealt with in this paper. The experimental results of a beam series tested at the Laboratory of Structures at Technical University of Madrid, Spain, are studied. Three beams of 2000 × 125 × 250 mm external dimensions have been tested, loaded in three-point bending scheme with a span length of 1600 mm between supports, as represented in Fig. 4. The cross-section of each beam follows the scheme analyzed in the example of Section 2, i.e. the tests have included a reference RC beam (RCS-U0) and two UHPFRC-RC beams, with a UHPFRC layer at the tensile side of 35 and 55 mm thickness (RCS-U35 and RCS-U55, respectively). The longitudinal reinforcement has consisted of 2 bars of 16 mm of B500SD steel, at the bottom and the top. The distance from the centroid of the steel bars to the lower edge of specimens was 58 mm. Stirrups of 8 mm diameter were placed at a distance of 150 mm to each other. The beams were loaded at the midspan under displacement-control at a rate of 0.02 mm/s. The midspan deflection was measured with a LVDT of 100 mm and the load applied with the hydraulic actuator was recorded with a built-in load cell. In addition, digital image correlation technique was used to record the deformations at one side of the specimens, as described in Section 3.3. The beams have been manufactured in the laboratory in upside down position. Firstly, the steel reinforcement was placed in the moulds. The conventional concrete, supplied by a local contractor, was

Average CoV

fCU [MPa]

ECU [MPa]

σcc [MPa]

εpc [–]

σpc [MPa]

GF [kN/ m]

135.4 0.02

53,382 0.03

4.6 0.07

0.0022 0.32

6.2 0.11

9.9 0.15

then poured until a total height of 250, 215 and 195 mm for RCS-U0, RCS-U35 and RCS-U55, respectively, thereby leaving empty the place for the UHPFRC layer. The free surface of conventional concrete was not treated at this stage, leaving its natural roughness. The main properties of the conventional concrete are detailed in Table 1. The characteristic yield and ultimate strength of reinforcing steel were 500 MPa and 575 MPa, respectively. The UHPFRC was manufactured and placed on the conventional concrete 15 months later. The mixture has consisted of a CRC premix kindly supplied by Hi-Con A/S (Denmark). The dry-mortar premix was mixed with the water and the fibers following the instructions of the supplier in a vertical pan mixer. A 2% volumetric amount of steel fibers of 12.5 mm length, 0.3 mm diameter and 2950 MPa yield strength were used. The fresh UHPFRC was poured on the conventional concrete surface, which was washed, slightly wirebrushed and kept moist. The stirrups were partially embedded by the UHPFRC in the RCS-U55 specimen. The beams were tested 2 months after the UHPFRC fabrication. During the storage, the beams were placed horizontally on the laboratory floor in upside down position.

250

2 Φ 16 LVDT (100 mm range)

UHPFRC layer

1600 2000

2 Φ 16

RCS-U0

2 Φ 16

125

RCS-U35

Φ 8 @ 150

55

2 Φ 16

2 Φ 16

Φ 8 @ 150

35

250

Φ 8 @ 150

125

Cylinder indirect tensile strength: 2.3 MPa (testing age)

UHPFRC (properties at testing age)

2 Φ 16

2 Φ 16

Water/Cement ratio: 0.5

125

RCS-U55

Fig. 4. Test set-up and instrumentation. Dimensions in mm. 5

2 Φ 16

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120

The material properties of the UHPFRC were determined from cubic (150 mm side) and prismatic specimens (100 × 100 × 500 mm). Six cubic samples were used to obtain the compressive strength (refer to Table 1). Six prismatic samples were employed to determine the tensile properties in flexural tests, in unnotched 4-point-bending tests (4 samples) and notched 3-point-bending tests (2 samples), both with a span length of 420 mm between supports. The results of bending tests are represented in Fig. 5. The average equivalent bending strength was 14.1 MPa. Multiple microcracking was observed in the central zone of 4-point-bent samples, with very thin cracks at 12–16 mm spacing, which closed after the formation of a macrocrack. The inverse method by [12] has been used to determine the tensile properties of UHPFRC from 4-point-bending tests (ECU, σcc, εpc, σpc in Table 1). The fracture energy (GF in Table 1) was obtained from the results of 3-point-bending tests. Additionally, the free shrinkage strain of UHPFRC was measured by 2 cylinders of 150 × 300 mm, which gave an average strain of 180 × 10−6 from manufacturing to testing age. Shrinkage was probably responsible for longitudinal cracks developed during the storage along the interface near the beam ends, with a length up to 200 mm and a width smaller than 0.1 mm.

Load, F [kN]

100 80 60 40 RCS RCS-U35 RCS-U55

20 0 0

5

10 15 20 25 30 Midspan deflection, v [mm]

35

(a)

RCS

3.2. Test results All three beams developed a flexure-governed failure. The loadmidspan deflection diagrams are represented in Fig. 6(a). Pictures of the crack pattern of tested specimens after testing are provided in Fig. 6(b). In all cases, spalling of the compression zone occurred when the reinforcing steel was well within the plastification stage. The reference RCS-U0 beam presented a typical bending response, with uncracked stage, formation and progressive development of vertical flexural cracks, yielding of the reinforcement, and spalling of the compression zone. The first vertical crack formed at a load of 10 kN and subsequent bending cracks progressively opened at a distance of about 150 mm to each other. It has to be noted that 150 mm was the spacing of stirrups. From a load of about 60 kN, very thin diagonal cracks also formed, as can be observed in Fig. 6(b). The presence of stirrups was fundamental to avoid shear failure, which did occur in similar specimens without shear reinforcement tested by the authors [13]. The specimens with UHPFRC layer presented a very similar global load-

40

Total test

RCS-U35

RCS-U55

(b) Fig. 6. Experimental results: (a) Load-midspan deflection diagrams; (b) Crack patterns at failure.

midspan deflection response to the reference RC beam (Fig. 6(a)), but remarkable differences can be observed by focusing on the pre-peak stage (Fig. 7). The initial response of specimens with UHPFRC layer started to differ from that of the reference RC beam from the cracking load of 10 kN, when the stiffness of RCS-U35 and RCS-U55 did not

25

15

5

10 v [mm]

40

0.5

1.0 v [mm]

F [kN] F [kN]

F [kN]

10

5

5 0

0 0.0

10

0 0.0

1.5

[MPa]

0

15

eq

10

Test detail

15

[MPa]

5

10

20 eq

20

5 w [mm]

25

30 10

0 0

15

Test detail 15

[MPa]

0

5

5 0

0

0

10

eq

5

10

10

15

[MPa]

10

20

eq

F [kN]

30

15

Total test

20

0.5 w [mm]

1.0

(b)

(a)

Fig. 5. Results of material tests of UHPFRC: (a) Load-midspan deflection of 4-point-bending tests; (b) Load-crack width of 3-point-bending tests. 6

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60

F = 50 kN v = 3.3 mm

Load, F [kN]

50 2

40

F = 82 kN v = 6.0 mm

30 1

20

F = 108 kN v = 10.3 mm

RCS RCS-U35 RCS-U55

10 0 0

1

2

3

F = 100 kN v = 24.0 mm

4

Midspan deflection, v [mm] Fig. 7. Pre-peak stage of load-midspan deflection diagrams.

(a) 5

experiment observable decrease. A slight stiffness decrease took place at a load of 12–14 kN for RCS-U35 and RCS-U55, which was due to the matrix microcracking (point ① as referred to in Section 2). That load corresponds to a bending moment of 4.8–5.6 m⋅kN, which agrees rather well with point ① of Fig. 3. Thereafter, Fig. 7 shows that a thicker UHPFRC layer resulted in a higher global stiffness. An observable stiffness decrease took place at a load of 22 and 32 kN for RCS-U35 and RCS-U55, respectively, which was due to opening of a first macrocrack at the UHPFRC. Such macrocrack formation was also detected by DIC, as described in Section 3.3. The second stiffness decrease corresponds to point ② as defined in Section 2. The bending moment corresponding to the formation of the first macrocrack was 8.8 and 12.8 m⋅kN for specimens RCS-U35 and RCS-U55, which is much smaller than the prediction of the model in Section 2 (Fig. 3). Such results indicate that the hardening capacity of the UHPFRC of the beams was poorer than the one expected from the properties obtained in material tests (Table 1). This observation has been noted by Habel [9] too, and it is further discussed in Section 5 of this paper. Beyond first macrocracking, further macrocraks formed at a distance of about 150 mm, in agreement with the stirrup spacing. According to the experimental results, the response of RCS-U35 and RCS-U55 approached rather rapidly the behavior of RCS-U0, without significant contribution of the UHPFRC when the longitudinal reinforcement yielded. Furthermore, the specimens with UHPFRC showed interlayer vertical debonding in the shear spans from a previously formed bending crack, as it can be observed from the crack patterns of Fig. 6(b).

0 Pre-peak (F = 82 kN)

v [mm]

-5 -10

Post-peak (v = 16 mm)

-15 -20 0

500

1000 x [mm]

1500

2000

(b) Fig. 8. DIC analysis of test RCS-U0: (a) Crack pattern; (b) Vertical displacements.

shape, cracks formation and width, and relative displacements at the interface between UHPFRC and RC. Strains are not used in the analysis, as the resolution of the strain field is not very accurate, though strain contours can provide interesting pictures of the crack pattern. The DIC analysis of test RCS-U0 is summarized in Fig. 8. The evolution of the crack pattern can be followed by the contours of horizontal strains. The formation of bending cracks at a constant distance to each other is observed in the graphic corresponding to service load (F = 50 kN) in Fig. 8(a). The formation of thin diagonal cracks at higher loads can be also noted in Fig. 8(a) for F = 82 kN. The other contours corresponding to the yielding stage show the formation of failure at the midspan, including spalling of the compression zone. In addition, the analysis of vertical displacements along the beam axis (Fig. 8(b)) shows the change of deformed shape, from parabolic function before yielding to triangular form upon the formation of plastic hinge at midspan. The DIC analysis of tests RCS-U35 and RCS-U55 has focused on the crack evolution and the interlayer interaction, so that it can be useful to understand the bending behavior of UHPFRC-RC elements. The horizontal displacements (u) along two horizontal lines at the level of the longitudinal reinforcement and the center of the UHPFRC layer are represented in Fig. 9(a) for test RCS-U35. From these graphics, the position of cracks can be detected by a jump of u, and the width of the crack is given by that jump. The study of the evolution of u shows first formation of 3 UHPFRC macrocracks under a load of F = 22 kN (in agreement with point ① in Fig. 7), at x = 661 mm, 932 mm and 1231 mm, without being visible at the RC. At a load of F = 29 kN, an additional UHPFRC macrocrack opens at x = 1030 mm and 4 cracks are present of the RC. All but one RC crack coincide with the position of UHPFRC cracks. The one which does not coincide is placed at x = 769 mm at RC and x = 661 mm at UHPFRC. Further load increases lead to more cracks at both layers UHPFRC and RC: under F = 48 kN,

3.3. Digital image correlation (DIC) Digital image correlation (DIC) is an experimental technique able to follow the deformed pattern of 2-D or 3-D problems from the photogrammetric evolution of a speckle pattern painted on a solid. In the present study, a speckle pattern was painted on one of the sides of tested specimens in order to follow the in-plane deformations and cracking process. A Nikon D90 camera with lens Nikon 18–200 mm f/ 3.5–5.6 with a resolution of 4288 × 2848 px was placed perpendicularly in front of the midspan of the painted side. Pictures were taken at a rate of 0.2 fps with a focal length of the lens of 24 mm. According to the camera resolution, 1 px corresponded to 0.5 mm approximately. The speckle pattern has consisted on randomly-painted black points of 1–3 mm size over a white background. The GOM Correlate software [14] has been used for the DIC analysis, with a facet size of 30 px and a point distance of 20 px. The main result of DIC technique is the displacement field of the analyzed surface. Strains and other parameters can be then also obtained by post-processing of displacements. In the present paper, the main attention is paid at displacements, which can provide very interesting results regarding the specimens’ deformed 7

Engineering Structures xxx (xxxx) xxxx

C. Zanuy and G.S.D. Ulzurrun

0.5

1.2 F = 22 kN, v = 1.2 mm

0.8 0.6

u [mm]

u [mm]

0.3 0.2

0.4

RC

0.1 0 0 1

500

1000 x [mm]

1500

UHPFRC

0

2000

0 1.6

F = 29 kN, v = 1.8 mm

1.2

0.6

0.8

u [mm]

0.8

0.4

500

1000 x [mm]

RC

0

UHPFRC

0

1.6 1.2

1500

2000

0

u [mm]

1 0 -0.5

0.2

-1

0

-1.5

1000 x [mm]

1500

2000

RC UHPFRC

0

3

F = 89 kN, v = 7.3 mm

500

1000 x [mm]

1500

2000

F = 107 kN, v = 9.3 mm

2 RC UHPFRC

u [mm]

1 0 RC

-1

UHPFRC

-2

0

500

1000 x [mm]

1500

0

2000

4

4

F = 105 kN, v = 11 mm

3

500

1000 x [mm]

1500

2000

F = 100 kN,. v = 21.3 mm

2 0

2 1

u [mm]

u [mm]

2000

0.5

0.4

3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5

1500

1.5

0.6

500

1000 x [mm]

F = 86 kN, v = 7.3 mm

2

0.8

0

500

2.5

RC UHPFRC

1 u [mm]

1000 x [mm]

-0.4

F = 48 kN, v = 3.6 mm

1.4

u [mm]

500

2000

F = 50 kN, v = 3.2 mm

UHPFRC

0

1500

0.4

RC

0.2

RC

0.2

UHPFRC

u [mm]

F = 32 kN, v = 1.3 mm

1

0.4

0 -1 -3 500

1000 x [mm]

1500

-6 RC UHPFRC

-10

UHPFRC

0

-4 -8

RC

-2

-2

-12

2000

0

(a)

500

1000 x [mm]

1500

2000

(b)

Fig. 9. Evolution of horizontal displacements along the longitudinal steel (RC) and the middle of the UHPFRC: (a) RCS-U35; (b) RCS-U55.

by the difference of vertical displacements between UHPFRC and RC in Fig. 10(a). Similar debonding is detected at a load of F = 89 kN around the crack at x = 1231 mm. The global behavior thereafter, especially upon begin of steel yielding, is characterized by debonding around

the number of cracks is 6 at the UHPFRC and 7 at the RC. It is interesting to note that the UHPFRC crack at x = 769 mm is surrounded by 2 cracks at the RC (x = 624 mm and 769 mm), which is associated to vertical interface debonding. The interface debonding can be analyzed 8

Engineering Structures xxx (xxxx) xxxx

C. Zanuy and G.S.D. Ulzurrun

0

x [mm] 1000

500

1500

As, Us

2000

hU

v [mm]

-2

F = 48 kN

x= 0

b

-4

sr/2

F = 89 kN

-6

σc

-8 -10

RC UHPFRC

F = 105 kN

-12

0 -2 -4 -6 -8 -10 -12 -14 -16 -18

500

x [mm] 1000

σc + dσc τb

σs

(a) 0

v [mm]

σc,cr = 0 σs,cr σU,cr x = sr /2

hc,eff

0

σs + dσs τb

1500

σc

2000

σc + dσc τU

σU

F = 50 kN

σU + dσU dx

F = 86 kN

Fig. 11. Composite tension chord and equilibrium of stresses.

contribution of concrete in tension between cracks (tension stiffening) is fundamental for the calculation of crack width and average curvature between adjacent cracks. The tension stiffening contribution can be accounted for by the study of the tension chord [15,16]. For UHPFRCRC sections, a composite tension chord model is necessary in order to include the contribution of UHPFRC and the RC-UHPFRC interaction. Therefore, a model is proposed in the following Section.

F = 107 kN

RC UHPFRC

(b) Fig. 10. Evolution of vertical displacements along the longitudinal steel (RC) and the middle of the UHPFRC: (a) RCS-U35; (b) RCS-U55.

those 2 UHPFRC cracks (see F = 105 kN in Figs. 9(a) and 10(a)), though the beam is able to carry loads until destruction of the compression zone. A similar analysis is carried out for RCS-U55 in Figs. 9(b) and 10(b). Two first macrocracks at UHPFRC formed at a load of F = 32 kN at x = 750 mm and 1240 mm. Those 2 cracks had associated debonding zone from a load of F = 50 kN. In total, 8 cracks subsequently formed at the UHPFRC and RC, though the width of UHPFRC cracks was rather small except for the cracks associated to debonding (see Figs. 9(b) and 10(b) at F = 86 kN and F = 87 kN). The collapse took place in the stage of plastification of longitudinal steel more prematurely than the other beams, probably due to the higher equivalent reinforcement ratio provided by both the longitudinal steel bars and the 55 mm thick UHPFRC layer. According to the former discussion, it can be concluded that DIC analysis is a powerful tool to understand the behavior of UHPFRC-RC beams, including the evolution of the interface. Probably, shearbending moment interaction is responsible for the vertical debonding as it has occurred in the shear spans. The lack of additional anchorage details at the end zones (e.g. slightly thicker UHPFRC layer) might have played a role too. As the present paper focuses on the bending behavior, the distortion produced by debonding will not be treated in the following Sections, but it will be a future research area. The analysis of the following sections will take advantage of the crack width records obtained with the DIC in order to compare the experimental bending moment-crack width diagrams with the model capabilities.

4.2. Composite tension chord model A composite tension chord model including the reinforcing steelconcrete and concrete-UHPFRC interactions is proposed. The model is focused on load levels larger than the one producing a macrocrack in the UHPFRC (point ②, as defined in Section 2). For smaller loads, the UHPFRC is characterized by strains and compatibility between the UHPFRC and the RC is straightforward, as shown in Section 2. A composite tension chord consisting of a steel area As (and perimeter Us) embedded into a concrete effective area of b × hc,eff, with a UHPFRC layer of b × hU bonded to the concrete, is considered, as plotted in Fig. 11. The effect of the curvature is neglected in this section, i.e. the tension chord is subjected to uniform tensile deformation and there are no stress/strain gradients. The influence of the curvature is re-introduced in Section 4.3. The portion between two adjacent cracks at a distance of sr to each other is analyzed. Due to symmetry, only one half of the crack spacing is studied (Fig. 11). At the crack (x = sr/2), the UHPFRC and the concrete have different crack widths (w and wU, respectively), which are to be estimated with the model. At the crack, the externally applied load (F) is carried by the UHPFRC (in softening) and the steel, while the concrete cannot carry tensile stresses:

F = σs, cr As + σU , cr bhU

(11)

Due to the bond stresses at the steel-concrete and concrete-UHPFRC interfaces, a fraction of the load is transferred from the steel and the UHPFRC to the concrete between cracks, thereby leading to tension stiffening contribution. The equilibrium of stresses at each material within a portion of length dx is represented in Fig. 11. The equilibrium equations at the steel, UHPFRC and concrete can be written as follows, respectively:

4. Proposed model 4.1. Introduction The bending behaviour of UHPFRC-RC members is a result of the contribution of concrete, longitudinal steel and UHPFRC, as well as the interaction between them. For conventional RC sections, the

dσs U dσ 4τ = τb s ⇒ s = b dx As dx Φ 9

(12)

Engineering Structures xxx (xxxx) xxxx

C. Zanuy and G.S.D. Ulzurrun

dσU τ = U dx hU

(13)

dσc Us 1 dσ 4τ τ = −τb − τU ⇒ c = − bρ − U dx bhc, eff hc, eff dx Φ hc, eff

(14)

width at the RC layer can then be derived as:

The maximum crack spacing can be also estimated by the model by limiting the maximum concrete stress at the midsection of the composite tension chord:

where τb and τU are the bond stresses between the steel and the concrete and between the concrete and the UHPFRC, respectively. If Eqs. (12) and (13) are introduced into Eq. (14), the transfer of stresses from the steel and the UHPFRC to the concrete can be calculated:

dσc dσ dσ = −ρ s − ρU U dx dx dx

σc, mid =

(15)

(16)

where τb0 = 2fct and τb1 = fct [15]. For the concrete-UHPFRC interface, τU = 0.2fct is assumed. The tensile strength fct is the one of the weakest layer, i.e. the conventional concrete. From the above equilibrium equations, the distribution of stresses between cracks can be represented (Fig. 12). The distribution of strains can be then calculated from stresses and the material laws. For the concrete, linear elastic behaviour is considered between cracks, i.e. the strain distribution results as plotted in Fig. 13. Thus, the average concrete strain leads:

εcm =

τU ρU sr τb ρsr + Ec Φ 4Ec hU

(17)

2hU

sr/2

σs

σs,cr σs,mid= σs,cr-2τbsr/Φ

x

σc σc,mid = 2ρ τbsr/Φ + +ρU τU sr/2hU

(18)

(19)

x

σU

(20)

The crack width of the UHPFRC is related with the stress at the crack by the softening law Eqs. (5) and (6), summarized as:

wU = wU (σs, cr )

τ U ρU

σs,cr σU,cr

And the elongation of the UHPFRC is due to the strains between cracks and the width of the crack:

ΔLU = εUm sr + wU

+

(23)

To extrapolate the composite tension chord model of Section 4.2 to members in bending, the effect of the curvature has to be taken into account. The model applies from the localization of a macrocrack in the UHPFRC. From that instant, the compatibility condition of Eq. (2) is no longer valid for hc ≤ y ≤ h, but it is replaced by the compatibility of extension concept of Eq. (18). The elongation at the level of the steel reinforcement can be calculated analogously to Eq. (19). From that value, the elongation at each depth ΔL(y) can be obtained by assuming

where the elongation of the RC can be calculated from the steel:

ΔLs = εsm sr

fct 2τ b ρ Φ

4.3. Cross-section under bending

For the steel, the elastic-plastic behaviour of Fig. 1(b) is assumed. Three situations can be obtained for the strain distribution, whether the steel has yielded or not (Fig. 14). The average steel strain between cracks can be derived as described in Appendix A. Regarding the UHPFRC, it has a different behaviour at the crack and in the sections between cracks. At the crack, the UHPFRC is in the softening stage, which means that its material law is given by Eqs. (5) and (6). Between cracks, the UHPFRC is characterized by strains, and the strain at each section can be obtained from the stress distribution of Fig. 12 and Eqs. (3) and (4). Accordingly, some sections of the UHPFRC between cracks can be in the hardening stage, while others may remain in the initial elastic domain. Three situations for the strain distribution of the UHPFRC may arise, as represented in Fig. 15. The average UHPFRC strain is calculated in Appendix B. A compatibility condition is expressed in terms of extensions. The total elongation of the composite tension chord (ΔL) has to be the same without regard of the fact that it can be calculated in the RC or in the UHPFRC. Thus:

ΔL = ΔLs = ΔLU

τU ρU sr ,max 2τb ρsr ,max + ⩽ fct ⇒ sr ,max ⩽ Φ 2hU

Nevertheless, the estimation of the crack spacing is not a goal of the present research, as its value is taken from experimental results in the analyses. In order to understand the capabilities of the composite tension chord model, explanatory examples have been completed and the results are represented in Fig. 16. The examples correspond to the geometric configuration and material properties of test RCS-U35 described in Section 3. For the concrete in tension, the effective area is estimated according to the Model Code [10]. As it was concluded in Section 3 that the hardening capacity of the UHPFRC of the beams has been smaller than the one obtained in material tests, the analysis has been done with two values of εpc (1.2‰ and 2.2‰). The graphics show that cracking takes place at the RC before the UHPFRC. Moreover, after macrocracking of UHPFRC, its width wU widens with load increases and approaches the width of RC.

where ρ = As/(bhc,eff) and ρU = hU/hc,eff. Similarly to the tension chord model by Marti [15], it is assumed that bond stresses are uniformly distributed between cracks, i.e. a rigidplastic bond-slip model is used. Different values are adopted for τb if the steel has yielded or not, in agreement with Marti:

τb0, εs ⩽ εy τb = ⎧ ⎨ ⎩ τb1, εs > εy

(22)

w = (εsm − εcm ) sr

σU,cr σU,mid = σU,cr -τU sr/2hU

(21)

x

The above equations allow solving the composite tension chord model for any applied load F. There are three main equations ((11), (18) and (21)) with three unknowns (σs,cr, σU,cr and wU). The crack

Fig. 12. Distribution of stresses in the composite tension chord. 10

Engineering Structures xxx (xxxx) xxxx

C. Zanuy and G.S.D. Ulzurrun

c

For UHPFRC-RC specimens, the experimental results corresponding to the various cracks detected with the DIC have been plotted, but it is clear that some of the cracks cannot be represented by a pure bending model. In particular, the cracks around which vertical debonding developed are affected by shear-bending moment interaction, which produces a significantly higher crack width, as it can be observed from the experimental curves. Those cracks, indicated in Fig. 17(b) and (c), are not the focus of the model. For the cracks developed at the midspan (x = 932 mm and 1030 mm for RCS-U35 and x = 840 mm and 1078 mm for RCS-U55), to which the model applies, the comparison of experimental and model results is very good. It has to be noted that it has been necessary to reduce the hardening capacity of the UHPFRC to εpc = 1.2‰ in order to better reproduce the experimental results. Poorer properties of the UHPFRC of the beams than the one obtained in material tests were also observed by Habel [9]. The effect of curvature is a notable difference with respect to tension members: there is a load level from which wU becomes larger than w, because the UHPFRC is at a larger distance from the neutral axis than the steel reinforcement. In the experiments, such result is especially visible for RCS-U35. The results for RCS-U55 are excellent for RC cracks, but not so good for UHPFRC crack widths, probably due to the influence of the extension of the debonding zones. In general, it can be concluded that the model is able to calculate crack widths at both the RC and the UHPFRC with good accuracy. In order to complete the experimental verification, results of composite beams tested by Pimentel & Nunes [17] are analyzed too, in particular the specimens referred to as LSA and LSA-U. The geometry and test configuration are represented in Fig. 18(a). The expected material properties (tested values were not reported) are also included in the figure. The experimental results are presented in terms of bending moment-curvature diagrams. The curvatures were derived from LVDT measurements along a 750 mm distance over the intermediate support. The comparison with model results is drawn in Fig. 18(b). In the model, the experimental value of the crack spacing has been introduced. As in the previous analysis, it has been necessary to adapt the material properties of the UHPFRC to better reproduce the experimental response. The values adopted are indicated in Fig. 18(c). From the comparative graph of Fig. 18(b), it is shown that the model is able to reproduce very well the positive contribution provided by the UHPFRC layer with respect to the RC specimen and there is no need of a reference length.

c,mid= 2 bsr/Ec U U sr/2EchU cm

x Fig. 13. Distribution of concrete strains.

a linear extrapolation, as follows:

ΔL (y ) = ΔLs

y−x d−x

(24)

In addition, Eq. (20) is modified for each depth (y) due to the strain gradient:

ΔLU (y ) = εUm (y ) sr + wU (y )

(25)

And the compatibility equation is finally applied at each depth (y) by:

ΔL (y ) = ΔLU (y )

(26)

The composite tension chord model is solved at each position y of the cross-section for the unknowns (εc,sup, x) and the resulting distributions of σs,cr(y) and σU,cr(y) are introduced into the equilibrium Eqs. (7) and (8). Among the results of the model, the crack width distribution wU(y) is obtained. Because the analysis is done at the cracked section, an additional step is necessary to calculate the average curvature between adjacent cracks. Here, it is assumed that the neutral axis depth (x) remains constant between cracks, which allows to extrapolate the average curvature from the average steel strain as follows:

κm =

εsm ε = sm κ d−x εs, cr

(27)

5. Verification of model capabilities The capabilities of the bending model are examined by analyzing the results of tests described in Section 3. As it was shown from DIC analysis, the crack width can be extracted for each cross-section by the jumps of the horizontal displacements (Fig. 9). Accordingly, the experimental bending moment-crack width diagrams can be derived by the DIC technique and they can be compared with the model results. The comparison is represented in Fig. 17. The crack width is studied at the level of the reinforcement and at the center of the UHPFRC layer. For RCS-U0, the comparison is made at the level of the steel reinforcement only. The study of test RCS-U0 is included just for the sake of verification of the DIC technology to measure crack widths, as the model for RC beams is based on well-known background. The comparison between model and experiments (Fig. 17a) is very satisfactory. It is noted that a crack spacing of sr = 150 mm has been used in the models, as observed from the experiments. s

6. Conclusions From the study presented in the present paper, the following conclusions may be drawn: – Existing models for UHPFRC-RC in bending have some drawbacks, especially after the formation of a macrocrack at the UHPFRC, when the strain compatibility equation and the concept of the reference length fail to be generally applicable. – Experimental results have shown that the bending response of UHPFRC-RC elements can be divided into different stages according

ls

s s,cr sm

=

s,mid

s,cr

s,cr-2 bsr/E s

(A)

s,cr /Es

s,cr -f y)/E p

f y/Es

sm

s,mid

x

fy/Es+(

ly

s

fy/Es+(

s,mid -f y)/E p

(B)

x

Fig. 14. Distribution of steel strains. 11

=

s,mid

s,cr

+(

s,cr-2 bsr/E s

(C)

x

f y/Es+

s,cr -f y)/E p

Engineering Structures xxx (xxxx) xxxx

C. Zanuy and G.S.D. Ulzurrun

U U,cr

+(

lcc

U

cc /EcU +

U,cr

U,cr - cc )/E pc

U,cr - cc)/E pc-

U,mid

U,cr /EcU - U sr /2EcU hU

x

(B)

cc /EcU + U,cr - cc )/E pc

U,cr

cc /Ecu

U,mid

x

(A)

l pc

U

Um

Um cc/EcU +( U,mid - U sr/2E pchU

U,cr /EcU

+(

U,cr /EcU - U sr /2EcU hU

x

(C)

300

300

250

250

Tensile load, F [kN]

Tensile load, F [kN]

Fig. 15. Distribution of UHPFRC strains.

200 150

100

Concrete UHPFRC

50 0 0.0

0.1

0.2

0.3

0.4

50

300

250

250

200 150 Concrete UHPFRC

50 0.0

0.1

0.2

0.3

8.0E-04

1.6E-03

2.4E-03

Average strain [-]

300

100

Steel UHPFRC Concrete

100

Crack width [mm]

0

(b)

150

0 0.0E+00

0.5

Tensile load, F [kN]

Tensile load, F [kN]

(a)

200

0.4

200 150

100 50 0 0.0E+00

0.5

Crack width [mm]

Steel UHPFRC Concrete 8.0E-04

1.6E-03

2.4E-03

Average strain [-]

Fig. 16. Example of application of composite tension chord model to tests RCS-U35: (a) εpc = 1.2‰; (b) εpc = 2.2‰.

estimate correctly the bending response including the tension stiffening effect. – It is shown that the aforementioned compatibility condition formulated in terms of extensions provides more rational results than the plain strain condition with reference length. So, the model allows overcoming the issues of the reference length concept. The average curvature between adjacent cracks and the crack widths at the RC and UHPFRC levels can be calculated with the proposed mechanically sound model.

to the tensile properties of the UHPFRC. The DIC technique has allowed understanding the formation of cracks at the UHPFRC and the RC, as well as to detect debonding between both layers in zones of shear-moment interaction. The DIC analysis has allowed estimating crack widths at the various cross-sections of tested beams. – A new model is presented in the paper. The model for bending departs from a composite tension chord model which accounts for the interaction between the three materials (concrete, reinforcing steel and UHPFRC) and the stress transfer mechanism through bond stresses. The compatibility condition is established in terms of the extension of each layer RC and UHPFRC, rather than by a traditional strain compatibility with perfect bond assumption. The former is necessary since the deformation state of the UHPFRC cannot be defined in terms of strains upon macrocracking formation. Therefore, the extension of the UHPFRC has to be calculated from the contribution of the width at cracked sections (in softening stage) and the elongation of intermediate sections (in elastic and/or hardening stages), while the extension of the RC can be estimated by integration of steel strains between cracks. The model has shown to

Acknowledgements The financial support provided by the Spanish ministry for science, innovation and universities (Project ID BIA2016-74960-R AEI/FEDER, UE) is gratefully acknowledged. The CRC pre-mix and fibers were kindly provided by Hi-Con A/S (Denmark). Also thanks to “Fundación José Entrecanales Ibarra” for funding the PhD fellowship of the second author.

Appendix A. Equations to calculate the average steel strain between cracks in the composite tension chord According to the three situations plotted in Fig. 14: 12

Engineering Structures xxx (xxxx) xxxx

45 40 35 30 25 20 15 10 5 0

12

180

3

215

180

hU = 50 mm

250

Moment, M [m kN]

C. Zanuy and G.S.D. Ulzurrun

3

400

400

LSA-U

LSA

Model Test

0.0

0.2 0.4 0.6 0.8 Crack width, w [mm]

12

Applied load

1.0

700

Midspan cracks

(a) Debonding zone cracks

50

UHPFRC RC Model UHPFRC Model RC

0.0

0.2 0.4 0.6 0.8 Crack width, w [mm]

1.0

45 40 35 30 25 20 15 10 5 0

40 30

Crack Mean LSA LSA-U

20 10 0 0

(b) Moment, M [m kN]

1000

60

Moment, M [m kN]

Moment, M [m kN]

(a) 45 40 35 30 25 20 15 10 5 0

0.04

0.06

0.08

0.1

0.12

Curvature, [m -1 ]

(b)

Midspan cracks

Concrete: fc = 35 MPa, Steel: fy = 537 MPa UHPFRC (expected): UHPFRC (adopted):

= 6.7 MPa,

cc

= 6 MPa,

GPa, wc = 4.5 mm, w0 = 0.7 mm,

UHPFRC RC Model UHPFRC Model RC

0.2 0.4 0.6 0.8 Crack width, w [mm]

cc

pc

= 7.5 MPa,

pc

= 0.003, EcU =

45 GPa, wc = 4.5 mm

Debonding zone cracks

0.0

0.02

pc

= 7 MPa,

pc

= 0.001, EcU = 45

0 = 1.5 MPa

(c) Fig. 18. Study of beams tested by Pimentel & Nunes [17]: (a) Geometry and configuration; (b) Bending moment- curvature diagrams; (c) Material properties.

1.0

(c) Fig. 17. Comparison of experimental and model bending moment-crack width diagrams: (a) RCS-U0; (b) RCS-U35; (c) RCS-U55.

– Fig. 14(a). The steel remains in elastic domain along sr/2:

εsm =

εs, mid + εs, cr σs, cr τs = − b r 2 Es Es Φ

(28)

– Fig. 14(b). The steel has yielded at all sections:

εsm =

εs, mid + εs, cr 2

εs, mid = εy +

εs, cr = εy +

(29)

σs, mid − f y Ep

(30)

σs, cr − f y Ep

(31)

– Fig. 14(c). The steel has yielded at a distance ly next to the crack:

εsm =

εy 2

+

εs, mid ls + εs, cr l y (32)

sr 13

Engineering Structures xxx (xxxx) xxxx

C. Zanuy and G.S.D. Ulzurrun

ly =

σs, cr − f y 4τb

Φ

(33)

s ls = r − l y 2

(34)

Appendix B. Equations to calculate the average UHPFRC strain between cracks in the composite tension chord According to the three situations plotted in Fig. 15: – Fig. 15(a). The UHPFRC is in the hardening stage along sr/2:

εUm =

εU , mid + εU , cr σU , cr σ σ τ s = cc − cc + − U r 2 EcU Epc Epc 4Epc hU

(35)

– Fig. 15(b). The UHPFRC is in elastic domain along sr/2:

εUm =

εU , mid + εU , cr σU , cr τU sr = − 2 EcU 4EcU hU

(36)

– Fig. 15(c). The UHPFRC is hardening in a distance lpc next to the crack:

εUm =

lpc =

εU , cr lpc + εU , mid lcc σcc + 2EcU sr

(37)

σU , cr − σcc hU τU

(38)

s lcc = r − lpc 2

(39)

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