- Email: [email protected]
Splitting failure of precast prestressed concrete during the release of the prestressing force
Engineering Failure Analysis 16 (2009) 2618–2634
Contents lists available at ScienceDirect
Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Splitting failure of precast prestressed concrete during the release of the prestressing force J.C. Gálvez a,*, J.M. Benítez b, B. Tork c, M.J. Casati d, D.A. Cendón e a
Departamento de Ingeniería Civil: Construcción, E.T.S. de Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, 28040 Madrid, Spain Departamento de Vehículos Aeroespaciales, E.T.S. de Ingenieros Aeronáuticos, Universidad Politécnica de Madrid, 28040 Madrid, Spain Ain Shams University, El Cairo, Egypt d Departamento de Vehículos Aeroespaciales, E.U. Ingeniería Técnica Aeronáutica, Universidad Politécnica de Madrid, 28040 Madrid, Spain e Departamento de Ciencia de Materiales, Universidad Politécnica de Madrid, E.T.S. Ingenieros de Caminos, Canales y Puertos, C/Profesor Aranguren s/n, 28040 Madrid, Spain b c
a r t i c l e
i n f o
Article history: Received 23 February 2009 Accepted 10 April 2009 Available online 8 May 2009 Keywords: Concrete Prestressed concrete Precast concrete Bond-splitting Fracture
a b s t r a c t Bond between steel and concrete is fundamental for the transmission of stresses between both materials in precast prestressed concrete. Indented wires are used to improve the bond in these structural elements. The radial component of the prestressing force, increased by Poisson’s effect, may split the surrounding concrete, decreasing the wire confinement and diminishing the bonding. This work presents a testing procedure to obtain the bond–slip curves, between steel and concrete, during the releasing of the prestressing force. The experimental procedure allows study of the splitting failure of the concrete, induced by the action of the indented wire. The influence of the distance between wires, the thickness of concrete cover and the depth of the wire indentations on the bond and splitting are examined. Specimens with three depths of the wire indentations and three thicknesses of concrete cover were tested. Moreover, a numerical procedure is presented for modelling the bond–slip, taking into account the possible failure of concrete by splitting. The numerical procedure accurately reproduces the experimental records and improves knowledge of this complex process. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Precast prestressed concrete elements are widely used for construction in Europe. Nowadays, hollow core slabs and prestressed joists are profusely used for concrete structures in building construction. In both cases, the prestressing force is based on the bond between steel (wire or strand) and concrete. If attention is focussed on single wire, the estimated annual production in Europe of such single wire (4 and 5 mm diameter) for prestressed concrete exceeds 120,000 metric tons [1]. Single wire is annually used to cast more than 45 106 m of prestressed joists and about 15 106 m2 of hollow core slabs in Europe. The case of strands is more difficult to quantify, given that it is used for prestressed and post-tensioned concrete elements. Bond between prestressed steel and concrete is essential for the success of the prestressing system. Nevertheless, bond is a complex problem and depends on many parameters. Due to its significance for practical design, it has been investigated by technical committees, e.g. see Report FIB [2], with it being an attractive challenge for many researchers, as the long list of papers on the topic corroborates. * Corresponding author. E-mail addresses: [email protected] (J.C. Gálvez), [email protected] (J.M. Benítez), [email protected] (B. Tork), mariajesus. [email protected] (M.J. Casati), [email protected] (D.A. Cendón). 1350-6307/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2009.04.023
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
2619
Precast prestressed concrete structural elements are usually made as follows. The steel (strands or single wires) is prestressed in a large precast prestressing table by means of anchor heads. Such a table is usually larger than 100 m or even 200 m. The concrete is then cast and cured with an accelerated curing process. The prestressing force is released by approaching the anchor heads. Finally, the steel between individual structural elements is cut and the prestressing force transmitted to each one. The mechanisms that contribute to the bond between prestressed steel and surrounding concrete are chemical adhesion, friction and mechanical interlocking between wire indentations and concrete [3]. The geometry of the surface of the wire is extremely important for bond. Fig. 1 shows the indention geometry of the wire, and Table 1 shows the dimensions of the indentations. The interlock mechanism may be explained through an analogous mechanism such as that proposed by Tepfers [4,5] for bars in the reinforced concrete (see Fig. 2). When the prestressing force of steel is released at the end of the element by cutting the wire, it tends to pierce in the concrete and develops tangential stress (bond stress) and radial stresses at the interface between the steel and concrete. Both stresses may be related by means of the a angle (see Fig. 2). Furthermore, Hoyer’s effect is present at the ends, is directly related with Poisson’s effect. In the case of indented wires, the wedging action generated by Poisson’s effect is magnified by the indentations on the surface of the wire, increasing the tension ring on these local zones [6–9]. These aspects are beneficial for bond, though they may be harmful if concrete splits, dropping the confinement and diminishing the bond [6,7,10–13]. Bond behaviour of reinforced concrete has been examined by many researchers, such as Gambarova and Rosati [7], Abrishami and Mitchell [8], Tepfers and Olson [5], Ogura et al. [14], Jendele and Cervenka [15] and Malvar [16], among others. Less attention has been devoted to prestressed concrete elements. In this field, bond stress versus slip curves has been studied: den Uijl [10], Abrishami and Mitchell [17] and Tassios and Bonataki [18], with a certain amount of attention being paid to the splitting action of the radial stresses and loss of confinement induced by longitudinal cracking. Several researchers have adopted an analogy between the splitting action of the reinforcement in bond and the pressure of a liquid in a pipe or sleeve [19]. These are two dimensional plane strain models, which are focussed on concrete fracture with no relationship between steel sliding and radial stresses being included, with even Poisson’s effect being omitted [20,21]. This paper presents a test procedure to evaluate the bond-splitting when the prestressing force is transmitted by releasing the steel (wire or strand) in precast prestressed elements. The influence of the distance between wires, the thickness of the concrete cover and the depth of the wire indentations in the bond-splitting is studied. Twenty-seven specimens, combining three thicknesses of concrete cover and three depths of the wire indentations have been tested. This work also includes a numerical procedure to simulate the bond and the possible coupled splitting process. The bond model is conceptually based on the proposal of Cox and Hermann [22–24] for ribbed steel bars by means of an interface with a non-associative plasticity model. Our model is performed for prestressed concrete and only includes parameters that can be experimentally measured. The bond model is also a non-associative plasticity model incorporated in an interface finite element. This bond model is coupled with a fracture model for concrete to take into account the splitting of the concrete. The experimental aforementioned results are used to check the model. This work is not intended to be either a substituting or modelling of transmission length, or one that and evaluates bond strength by standardised testing methods. With this paper the authors seek to emphasise that with this new approach, it is possible to analyse the influence of the parameters affecting bond in the splitting failure of the concrete. It is acknowledged that further work should be carried out to extend this modelling to the majority of the prestressed concrete structural elements. Whereas the following section presents the experimental program, Section 3 examines the experimental results. Numerical procedure is then presented in Section 4, with the model validation being presented in Section 5. Discussion is presented in Section 6. Finally, the conclusions obtained from the model and the experiments are presented in Section 7. 2. Experimental programme Fig. 3 shows two engineering examples of the bond splitting failure in precast prestressed concrete joists. In both cases, the splitting failure, caused by the bond, was detected during the release of the prestressing force. It is worth noting that, due to Hoyer’s effect, the splitting effect is more marked in the edges of the joists (Fig. 3b).
Fig. 1. Geometry of the indentations of the wire [36].
2620
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
Table 1 Specified indentation dimensions, according to UNE 36094 [36]. Nominal wire diameter (mm)
Dimensions of indentations Depth a (mm 102) Type 1
3 4 5 6 P7
Length l (mm)
Spacing p (mm)
3.5 ± 0.5
5.5 ± 0.5
5.0 ± 0.5
8.0 ± 0.5
Type 2 2–6
3–7 4–8 5–10 6–12
5–9 6–10 8–13 10–20
Fig. 2. Bond behaviour of deformed bars, radial component of the bond stresses balanced by tensile stresses in the uncracked ring of concrete [4].
The experimental program was performed to analyse in laboratory, by a simplified modelling, the observed coupled bond-splitting process (Fig. 3a). Fig. 4 shows a sketch of the joist with the geometry and dimensions adopted for the specimens in the experimental analysis. For the sake of simplicity, specimens with only one wire and a rectangular shape cross section were performed. 2.1. Materials and specimens The specimens were manufactured with concrete, composed of Portland cement, siliceous sand and siliceous crushed coarse aggregates of 6 mm maximum size. Table 2 shows the composition of the mixture. Table 3 shows the mechanical properties of the concrete at the test age. EN-10138 Y 1770 C 4 I [25] wires were selected. Table 3 shows the mechanical properties of the wires, being the nominal diameter 4 mm. Wire with three indentation depths (shallow, medium and deep) were used. Table 4 shows the wire indentation depths. Twenty-seven prismatic specimens were cast with a wire embedded longitudinally in the specimen. The dimensions of the prismatic specimens were 400 mm in length, 60 mm in width, and with three different concrete covers. Fig. 5 shows the geometry and dimensions of the specimens. The cross section of the specimens had the shape of a slender rectangle to obtain only two splitting cracks, perpendicular to the longer sides of the rectangle. Table 5 shows the specimen nomenclature and the concrete cover. The thinnest cover was adopted in accordance with the smallest distance between wires in a common prestressed joist or hollow core slab. Before casting the concrete, the wires were tensioned to 17 kN in stiff vertical steel frames. The wire was fixed to the steel frame by wedge anchorages; the lower one was directly supported by the bottom plate, and the upper one fixed to a displaceable externally threaded piece (Fig. 6a). The wire was stressed in a servo-controlled testing machine. The prestressing force was applied monotonically up to the desired value. To keep the prestressing force in the wire, a U-shaped washer was placed between the upper plate and the displaceable anchorage, the nut of the threaded piece being tightened against the Ushaped washer. Then the testing machine was unloaded, transmitting the force to the prestressing frame, and this was then removed from the testing machine. The final prestressing stress of the wire was measured and controlled by a displacement extensometer placed on the central part of the wire. Fig. 6a also shows the safety device added as a protection against accidental snapping of the wire. Once the wires were prestressed, the specimens were cast horizontally in one layer in the ground steel moulds, jointed to the prestressing frames by screws. This screw device guaranteed the alignment of the wire with the longitudinal axis of the specimen and the horizontality during casting and vibrating. The concrete was compacted in the moulds on a vibrating table. The specimens were left in the moulds for 24 h, covered with saturated sacking at room temperature, and then covered with three coats of waterproof paint to maintain humidity during the curing process. The specimens were tested 28 days after casting. For detailed information on specimen preparation, see [26].
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
2621
Fig. 3. Two examples of splitting failure in precast prestressed joists.
2.2. Experimental procedure The tests were performed in two stages: first, the prestressing force was transmitted from the frame to the testing machine, and then the prestressing force from the testing machine to the concrete prism by the controlled release of the prestressing force of the wire. Fig. 7 shows a sketch of the prestressing frame, with the specimen in the testing machine. At the beginning of the first stage, the wire was stressed, the prestressing frame was compressed, and the concrete prism was unstressed. The prestressing frame was coupled to the testing frame of the machine. A tensile load was applied by displacing the machine actuator downward at a rate of 0.1 mm/min till the prestressing force (17 kN) was reached. When the force was completely transmitted to the testing machine, the prestressing frame and the concrete prism were unloaded. Finally, the nut of the threaded moving anchor was loosened and the U-shaped separator was taken off. In the second stage, the prestressing force of the wire was transferred, under control, to the concrete prism, by moving the actuator upward at a rate of 0.3 mm/min. The gradual release of the wire transferred the force to the concrete prism. The test ended when the free ends of the wire were completely unloaded. During the tests the following parameters were recorded: Released load supplied by the testing machine. Displacement of the actuator of the testing machine. Longitudinal shortening of the concrete prism, measured with a gage length of 387.5 mm. Measurements were taken on opposite faces and the mean value was recorded. Wire–concrete slip on the upper and lower faces of the prismatic specimens.
2622
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
Fig. 4. Sketch of a joist with a longitudinal splitting crack. The part selected for the study is indicated by a different weave.
Table 2 Composition of the concrete mixture. Component
Quantity (kg/m3)
Mix proportion by weight
Cement Water Sand Coarse aggregate
450 225 900 720
1.00 0.50 2.00 1.60
Table 3 Mechanical properties of the materials. Concrete
Steel wire
E = 38 GPa fc = 57 MPa ft = 3.0 MPa GF = 100 N/m
E = 226 GPa r0.2 = 1755 MPa ru = 1935 MPa eu = 5.25%
Table 4 Depth of the wire indentation. Denomination
Indentation depth (mm)
Shallow Medium Deep
0.01–0.02 0.04–0.06 0.1–0.11
Crack opening displacement of the longitudinal cracks. Measurements were taken on opposite faces, those with the thinnest cover. 2.3. Testing equipment The prestressing load was applied by a servocontrolled testing machine, and was measured with a 25 kN load cell with ±0.5% error at full scale. Extensometers, of ±2.5 mm nominal travel and ±0.15% error at full scale displacement, were used to measure the shortening of the specimen, the longitudinal crack opening displacement, and wire–concrete slip on the upper and lower faces of the prismatic specimen (see Fig. 8 for details of the placement of the extensometers). 3. Experimental results Fig. 9a–c shows the experimental records of released load versus longitudinal shortening of the concrete prism, for specimens with shallow, medium and deep wire indentations, and concrete covers of 5, 9 and 13 mm. The recorded curves of the specimens with the thinnest concrete cover show a point where the slope of the curves markedly changes. Behind this point the longitudinal shortening of the concrete prism diminishes while the testing machine releases the tension of the wire. This special point marks the beginning of the longitudinal splitting of the concrete prism. The splitting cracks reduce the friction
2623
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
(the wire coincides with the longitudinal axis of the concrete prism) Fig. 5. Geometry and dimensions of the specimens.
Table 5 Dimensions of the tested specimens (see Fig. 5). Specimen
Thickness of specimen (mm)
Concrete cover (mm)
C1 C2 C3
14 22 30
5 9 13
Fig. 6. (a) Detail of displaceable anchorage of the wire for prestressing and (b) horizontal steel mould attached to the frame with the prestressed wire.
2624
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
Fig. 7. Sketch of the test setup.
between the steel and concrete due to the loss of the wire confinement, and consequently induce a release of part of the compressive force in concrete, which diminishes the longitudinal shortening of the concrete prism, as seen in the experimental records. Specimens with 9 and 13 mm thickness concrete cover do not show this behaviour. Fig. 10 compares the experimental records of released load versus longitudinal shortening of the specimens with the thinnest concrete cover and shallow, medium and deep wire indentations. The deepest indentation gave the smallest load of concrete splitting, which suggests that deeper indentation leads to a more intensive tension ring in the concrete, owing to the mechanical interlock between the steel indentations and the concrete. The average splitting loads were 15, 13, 12.5 kN for specimens with wires of shallow, medium and deep indentations, respectively. Moreover, before concrete splits, the deepest the indentation the greatest shortening of the concrete prism. Fig. 11 shows the experimental records of the released load versus crack opening displacement (COD) of the splitting crack, measured on the upper face of the specimen (extensometer 3 in Fig. 8). Each curve shows a break point corresponding to the opening of the crack, shown by a fast rise of the extensometer measurement. The break point load coincides with that in Figs. 9a–c and 10, and confirms the interpretation of those figures. Fig. 12 shows the experimental records of released load versus penetration of the wire into concrete, measured on the upper face of the specimens (extensometer 2 in Fig. 8), for specimens with deep wire indentation, and covers of 5, 9 and 13 mm. Specimens with a concrete cover of 5 mm show a marked change in the slope of the curve. This point corresponds to the start of splitting and complements the above comments for these specimens. The splitting leads to a loss of confinement of the wire, increasing the slip between concrete and steel, with a greater penetration of the wire into concrete. Similar behaviour was observed in the specimens with shallow and medium depth wire indentation. Although specimens with shallow indentation showed a smoother change in the curves than ones with medium and deep indentation. Experimental curves of load versus penetration of the wire in the concrete, measured on the bottom face of the specimens, were practically equal to those of the upper face.
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
2625
Fig. 8. Extensometers and measure devices for: (1) longitudinal shortening, (2) penetration of the wire in the upper face of the concrete prism and (3) COD of the splitting cracks.
4. Numerical procedure 4.1. Problem posing Two processes have to be modelled: (1) the possible splitting of the concrete induced by the radial pressure of the wire and (2) the bond in the interface between concrete and steel. Both processes are related. In this work the model of the splitting concrete failure is based on the cohesive crack approach [27], and that of the bond on a plasticity formulation [26,28]. Fig. 13 shows a sketch of both processes to be modelled. 4.2. The cohesive crack model The cohesive crack model is generally accepted as a realistic simplification of the fracture of quasi-brittle materials. Such a model was proposed by Hillerborg et al. [27] in the late seventies, and has been successful in the analysis of the fracture of concrete and concrete-like materials since its proposal. The softening function, r = f(w), is the main ingredient of the cohesive crack model. This function, a material property, relates the stress r acting across the crack faces to the corresponding crack opening w (see Fig. 14). In mode I opening, the stress transferred, r, is normal to the crack faces. A detailed study of this model has been published by Bazant and Planas [29].
2626
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
a
16
Released load (kN)
14 12 10 8 6 Shallow indentation 4 2 0 -0.2
Cover=5 mm Cover=9 mm Cover=13mm
-0.15
-0.1
-0.05
0
Longitudinal shortening (mm)
b
16
Released load (kN)
14 12 10 8 6 Medium indentation 4 2 0 -0.2
Cover=5 mm Cover=9 mm Cover=13mm
-0.15
-0.1
-0.05
0
Longitudinal shortening (mm)
c
16
Released load (kN)
14 12 10 8 6 Deep indentation 4 2 0 -0.2
Cover=5 mm Cover=9 mm Cover=13mm
-0.15
-0.1
-0.05
0
Longitudinal shortening (mm) Fig. 9. Released load versus longitudinal shortening of the concrete prism in specimens with 5, 9 and 13 mm of thickness concrete cover and the wire indentation: (a) shallow, (b) medium and (c) deep.
Two properties of the softening curve are most important: the tensile strength, ft, and the cohesive fracture energy, GF. The tensile strength is the stress at which the crack is created and starts to open (f(0) = ft). The cohesive fracture energy, GF, also called specific fracture energy, is the external energy supply required to create a full break unit surface area of a cohesive crack, and coincides with the area under the softening function. The tensile strength and the specific fracture energy are material properties and may be experimentally measured in accordance with ASTM C 496 [30] and RILEM 50-FMC [31], respectively.
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
2627
16
Released load (kN)
14 12 10 8 6 Cover=5 mm
4
Shallow indentation Medium indentation Deep indentation
2 0 -0.2
-0.15 -0.1 -0.05 Longitudinal shortening (mm)
0
Fig. 10. Released load versus longitudinal shortening of the concrete prism in specimens with 5 mm of thickness concrete cover and three depths of the wire indentation.
16
Released load (kN)
14 12 10 8 6
Cover = 5 mm
4
Shallow indentation Medium indentation Deep indentation
2 0 0
0.05
0.1
0.15
0.2
0.25
COD (mm) Fig. 11. Released load versus COD of the splitting crack of the concrete prism in specimens with 5 mm of thickness concrete cover and three depths of the wire indentation.
4.3. The bond model The interface between concrete and steel is able to transmit normal and tangential stresses and shows dilatancy. The barscale approach, according Cox’s terminology [23,26], is adopted, then the wire–concrete interface is idealized in terms of two simplifications: (1) uniform distribution of the bond stress for a unit surface element (Fig. 15a), and (2) idealised deformation of the concrete in the bond zone (Fig. 15b). 4.3.1. Yield surface for bonding The relative displacement between the wire and concrete at the interface is vectorial in nature. We denote it as u, with normal and shear components denoted as un and ut, i.e.,
n þ ut~ u ¼ un~ t
ð1Þ
where ~ n and ~ t are the unit vectors, respectively, normal and tangential to the surface of the wire. Likewise, the stress transferred on the interface between the concrete and the wire is also vectorial, and is characterised by the vector t acting on the interface; with normal and tangential components denoted as r and s, i.e.,
n þ s~ t ¼ r~ t
ð2Þ
In this work, elastoplastic formulation is adopted, in which the interface displacement is split into its elastic and inelastic parts
u ¼ u e þ ui ;
u_ ¼ u_ e þ u_ i
ð3Þ
2628
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
16
Released load (kN)
14 12 10 8 6 Deep indentation
4
Cover=5 mm Cover=9 mm Cover=13mm
2 0 0
0.2
0.4 0.6 0.8 1 Wire penetration (mm)
1.2
1.4
Fig. 12. Released load versus wire penetration into concrete prism in specimens with 5, 9 and 13 mm of thickness concrete cover and deep depth of the wire indentation.
Fig. 13. Numerical problem processes: (a) longitudinal cracking of concrete (splitting) and (b) bond–slip behaviour of the interface between wire and concrete.
Fig. 14. Cohesive crack, softening function and notation for mode I fracture of quasi-brittle materials.
For the inelastic behaviour, it is assumed that the inelastic displacement can progress when the yield surface F(t) = 0 is reached. The following hyperbolic expression [32] has been assumed by the authors [33,34]:
FðtÞ ¼ s2 tan /f ðft rÞ 2c tan /f ðft þ rÞ
ð4Þ
where /f is the friction angle between the wire and concrete, c the cohesion (bond strength without interface normal stress), and ft the tensile strength in the normal direction to the interface. These values are instantaneous values and depend on loading history through the effective inelastic crack displacement uieff, defined by the conditions
0:5 u_ ieff ¼ ku_ i k ¼ u_ 2n þ u_ 2t Z uieff ¼ u_ ieff dt
ð5Þ ð6Þ
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
2629
Fig. 15. Sketch of the idealised interface between wire and concrete for a unit surface element: (a) uniform distribution of the bond stress and (b) deformation of the concrete in the bond zone.
In this work it is assumed that the friction angle /f is constant, while the instantaneous tensile strength ft and cohesion c depend on uieff bilinearly as depicted in Fig. 16. Following the analogy with the cohesive model, the area under the softening curve for ft may be interpreted as the specific fracture energy GIF for mode I, and the area under the softening curve for the cohesion c (also has dimensions of energy per unit area) as the mode IIa (shear under normal confinement) specific fracture energy GIIa F [35]. For the particular case of bond, this one may be measured by means of a pull-out test under high confinement. The following values for xc and xs in the softening curves (see Fig. 6) were adopted.
xc ¼ xr ¼
2GIIa F ðs1c þ c0 Þx1c s1c 2GIF
ðs1r þ ft0 Þx1r s1r
ð7Þ ð8Þ
Fig. 17 shows the plasticity surface for bonding in the interface, and its evolution with the parameter uieff. Note that for each state of damage, the hyperbolic plasticity surface has two branches and only the branch extending towards negative values of r is physically acceptable. Furthermore, it should also be noted that for fully damaged interface (complete loss of tensile strength and cohesion), the plasticity surface degenerates into a Coulomb friction surface with friction coefficient l = tan /f. 4.3.2. Flow rule and dilatancy Due to the dilatant behaviour of the wire-steel interface, non-associative plasticity has been adopted. Then, the evolution of the inelastic displacements in the interface zone is specified by means of the flow rule, given by
u_ i ¼ k_
@QðtÞ _ ¼ kb @t
ð9Þ
Fig. 16. Softening curves for the strength of the interface between wire and concrete: (a) normal direction (tensile strength) and (b) tangential direction (bond strength).
2630
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
Initial plasticity surface (u
ieff
= 0)
τ Intermediate plasticity surface (0 < u
φf
ieff
Final plasticity surface ieff
( u > wσ ) σ
Fig. 17. Plasticity surface and evolution.
where Q(t) is the plastic potential, b is the normal to the plastic potential surface and k_ is a non-negative plastic multiplier. The dilatancy angle is also defined
tan /d ¼
u_ in u_ it
ð10Þ
where /d is the dilatancy angle, u_ in and u_ it are the incremental inelastic displacements in normal and tangential directions to the interface, respectively. /d coincides with the angle formed by the potential surface and the negative part of the r axis (see Fig. 18). The dilatancy is also assumed to depend on the damage level through uieff. Following Ref. [35,36], a linear curve has been adopted:
8 < / 1 uieff 8uieff < ucd d0 ucd /d ¼ : 0 8uieff P ucd
ð11Þ
where /d0 is the initial value of the dilatancy angle and ucd is the critical inelastic crack displacement after which the interface ceases to exhibit the dilatancy effect.
φ
d
b
B φ
d
A φ
d
b
O
φ
d
A' φ
d
φ
d
b
B'
Fig. 18. Return direction to the plasticity surface of the inelastic corrector.
2631
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
When traction stress r is predominant over the tangential stress s, the direction normal to the plastic potential (Q = cte) is not defined. Consequently, in this case the return direction to the origin of stresses is adopted, which divides the stresses space into two parts, as is shown in Fig. 18. It should be pointed out that this is a non-associative plasticity approach, where normal directions to cracking surface and plastic potential are different. 4.3.3. Incorporation into a finite element code An interface element was developed to incorporate the bond model in the finite element code ABAQUSÓ. The cohesive model for simulation of the radial cracks was included by means of non-linear springs. Details regarding the finite element implementation are found in [26]. An arc length algorithm was adopted for the computational procedure and no special difficulties were found in achieving a convergent solution. Fig. 19 shows a sketch of the finite element mesh. To simplify the computations, the bulk behaviour of the material outside of the crack and the interface was assumed to be linear-elastic and isotropic, although this approximation could have been relaxed if necessary. 5. Model validation The presented numerical procedure is used to reproduce the experimental results. Table 3 shows the mechanical properties of the materials, and Table 6 the parameters adopted for the interface for the three depths of the indentations. These parameters were not experimentally measured, but estimated. Fig. 20 shows the experimental records and the numerical prediction of released load versus longitudinal shortening of the concrete prism, for specimens with shallow, medium and deep wire indentations, and the minimum concrete cover (5 mm). Fig. 21 compares the experimental records and the numerical prediction of released load versus COD of the possible splitting crack on the concrete prism, for specimens with shallow, medium and deep wire indentations, and the minimum concrete cover (5 mm). Fig. 22 shows the experimental records and the numerical prediction of released load versus wire penetration on the edge of a concrete prism, for specimens with medium wire indentations, and the minimum concrete cover (5 mm). In all cases the numerical prediction properly predicts the bond behaviour and the coupled splitting failure. The prediction fit quite well the entire behaviour of the experimental records. 6. Discussion There is agreement that, in reinforced concrete with ribbed bars, the splitting failure occurs when the concrete cover is lower than three times bar diameter [12]. This agreement is based on experimental pull-out results with ribbed bars, and analytical and numerical studies of the cross section of the element with an internal pressure on the bar hollow. However, there is not a detailed study for the bond-splitting on precast prestressed concrete, taking into account the bond parameters and their role in the splitting action. The proposed experimental procedure allows for study of the splitting failure and the bond in the interface steel-concrete as combined processes. This work concerns itself only with bond-splitting failure and is not intended to be a substitute of any method of bond strength or transmission length in prestressed concrete. The experimental results of the specimens with the thinnest concrete cover have shown that the depth of the indentations is crucial as regards the possible splitting failure. The smallest thickness of the concrete cover, 5 mm (1.25 times wire
Fig. 19. Sketch of the finite element mesh used for the numerical analysis.
Table 6 Parameters of the interface adopted for numerical modelling. Denomination
smax (MPa)
C0 (MPa)
/d (°)
/f (°)
Shallow Medium Deep
10 10 10
65.3 64.2 63.8
0.3 4 16
40 42 43
2632
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
a
16
Released load (kN)
14 12 10 8 6 4 2 0 -0.2
Shallow indentation Cover = 5mm Experimental Numerical
-0.15
-0.1
-0.05
0
Longitudinal shortening (mm)
b
16
Released load (kN)
14 12 10 8 6 4 2 0 -0.2
c
Medium indentation Cover = 5 mm Experimental Numerical
-0.15 -0.1 -0.05 Longitudinal shortening (mm)
0
16
Released load (kN)
14 12 10 8 6 4 2 0 -0.2
Deep indentation Cover = 5 mm Experimental Numerical
-0.15 -0.1 -0.05 Longitudinal shortening (mm)
0
Fig. 20. Experimental records and numerical prediction of released load versus longitudinal shortening of the concrete prism in specimens with 5 mm of thickness concrete cover and wire indentation: (a) shallow, (b) medium and (c) deep.
diameter), is apparently much thinner, but is based on the real dimensions used by industry for some of these precast prestressed joists (see Fig. 3). The comparison of the experimental records of the released load versus specimen shortening, COD and wire penetration has shown that the deepest indentation the highest splitting action of the wire. The numerical modelling accurately reproduces the experimental records. The fracture cohesive model only includes parameters that have been measured by standardised methods. The bond model includes parameters that may to be experimentally measured. In this work, some parameters for bond modelling (see Table 6) have been estimated and not experimentally measured. These adopted values look according with the experimental results, and may be obtained by an inverse analysis, out the scope of this work.
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
2633
Shallow indentation
16 Medium indentation
Released load (kN)
14 12 Deep indentation
10 8 6
Cover = 5 mm
4
Numerical Experimental
2 0 0
0.04
0.08 COD (mm)
0.12
Fig. 21. Experimental records and numerical prediction of released load versus COD of the splitting crack of the concrete prism in specimens with 5 mm of thickness concrete cover and three depths of the wire indentation.
16
Released load (kN)
14 12 10 8 6
Medium indentation Cover = 5 mm
4
Experimental Numerical
2 0 0
0.2
0.4 0.6 0.8 Wire penetration (mm) Upper face
1
1.2
Fig. 22. Experimental records and numerical prediction of released load versus wire penetration into concrete prism in specimens with 5 mm of thickness concrete cover and medium depth of the wire indentation.
7. Conclusions and final comments A testing procedure to study the splitting failure of precast prestressed concrete elements during the release of the prestressing force has been proposed. The procedure allows studying the parameters affecting the bonding in the splitting failure of the concrete. Twenty-seven specimens, combining three thicknesses of concrete cover and three depths of the wire indentations, have been tested. The following remarks may be made from the experimental work: Splitting failure has been observed on the specimens with the thinnest concrete cover. This failure was observed by means of a change in the slope of the released load versus longitudinal shortening of the specimen recorded curves. The longitudinal shortening of the specimen decreased after splitting failure, indicating the loss of wire confinement. The critical released load that showed the splitting failure diminished when the depth of the wire indentations increased. The COD measurement has made easier to detect and follow the splitting failure. Moreover, it has been experimentally observed that the deepest depth wire indentation the earliest (load released) and largest COD in the longitudinal cracking failure. The deepest depth wire indentation the best bond between wire and concrete, until the splitting failure. This better behaviour is shown by a lower sliding of the wire at the end of the specimen and by a larger longitudinal shortening of the specimen for equal released load in the recorded experimental curves. It is worth noting that the deeper wire indentation leads to a higher splitting stresses and to a larger COD in the specimens. A numerical procedure for modelling the bond–slip at the interface of the wire and concrete has been proposed. The procedure takes into account the possible failure of concrete by the splitting action of the wire. The bond modelling is based on a
2634
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
non-associative plasticity approach, and the splitting concrete modelling is based on the cohesive crack approach for quasibrittle materials. The bond model was incorporated into an interface finite element. The cohesive crack model for the simulation of the radial cracks was included by means of non-linear springs. Both models were incorporated in the finite element code ABAQUSÓ. The numerical procedure has been contrasted with the above experimental results, and accurately reproduces the experimental records. The experimental work and the numerical procedure emphasise that the cohesive crack models, in combination with the bond modelling, are promising tools in the simulation of the splitting failure of the precast prestressed concrete structural elements. Further more work must to be carried out to extend this modelling to full-scale structural elements. Acknowledgements The authors gratefully acknowledge the financial support for the research provided by the Spanish Ministerio de Educación y Ciencia under Grants BIA-2005-09250-CO3-02 and BIA-2008-03523 and by the Ministerio de Fomento under Grants MFOM-2004/9 and MFOM-01/07. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
and consulted on June 2008. FIB. Bond of reinforcement in concrete. State of art report. FIB bulletin, vol. 10; 2000. Janney JR. Nature of bond in pretensioned prestressed concrete. ACI J 1954;25(9):717–36. Tepfers R. A theory of bond applied to overlapped tensile reinforcement slices for deformed bars, Chalmers University of Technology, Division of Concrete Structures, Götegorg, Sweden, Publication 73:2; 1973. Tepfers R, Olsson PÅ. Ring test for evaluation of bond properties of reinforcing bars. In: Bond in concrete: from research to practice, vol. 1, no. 1. Latvia, Riga: Riga Technical University and CEB; 1992. p. 89–99. Gálvez JC, Tork B, Planas J. Concrete splitting during the release of prestressing force in indented wires. In: Balázs G, Bartos P, Cairns J, Borosnyói A, editors. Bond in concrete; 2002. p. 95–102. Gambarova PG, Rosati GP. Bond and splitting in reinforced concrete: t-test results on bar pull-out. Mater Struct 1996;29:267–76. Abrishami H, Mitchell D. Simulation of uniform bond stress. ACI Mater J 1992;89(2):161–8. Gustavson R. Experimental studies of the bond response of three-wire strands and some influencing parameters. Mater Struct 2004;37:96–106. den Ujil JA. Bond and splitting action of prestressed strand. Bond in concrete: from research to practice, vol. 1, no. 2. Latvia, Riga: Riga Technical University and CEB; 1992. p. 79–88. Gambarova PG, Rosati GP, Zasso B. Steel-to-concrete bond after concrete splitting: constitutive laws and interface deterioration. Mater Struct 1989;22(131):347–56. Cairns J, Johns K. The splitting forces generated by bond. Mag Concr Res 1995;47:153–65. Benítez JM. Study of the interaction between prestressed wire and concrete during release of the prestressing force. Ph.D. Thesis, Univ. Castilla La Mancha; 2006 [in Spanish]. Ogura N, Bolander JE, Ichinose T. Analysis of bond splitting failure of deformed bars within structural concrete. Eng Struct 2008;30(2):428–35. Jendele L, Cervenka J. Finite element modeling of reinforcement with bond. Comput Struct 2006;84(28):1780–91. Malvar J. Bond of reinforcement under controlled confinement. ACI Mater J 1993;89:593–601. Abrishami H, Mitchell D. Bond characteristics of pretensioned strand. ACI Mater J 1993;90(3):228–35. Tassios T, Bonataki E. Experimental analysis of the bond behaviour of prestressed tendons. In: Bond in concrete: from research to practice, vol. 2, no. 2. Latvia, Riga: Riga Technical University and CEB; 1992. p. 29–37. Noghabai K. Discrete versus smeared versus element-embedded crack models on ring problems. J Eng Mech 1999;125(3):307–15. den Uijl JA, Bigaj AJ. A bond model for ribbed bars based on concrete confinement. Heron 1996;41(3):201–26. van der Veen C. Splitting failure of reinforced concrete at various temperatures. In: van Mier JGM, Rots JG, Baker A, editors. Fracture processes in concrete rock and ceramics. RILEM; 1991. Cox JV. Development of a plasticity bond model for reinforced concrete – theory and validation for monotonic applications. TR-2036-SHR. Naval Facilities Eng. Serv., Port Hueme, CA; 1994. Cox JV, Hermann R. Development of a plasticity bond model for steel reinforcement. Mech Cohes-Frict Mater 1998;3:155–80. Cox JV, Hermann R. Validation of a plasticity bond model for steel reinforcement. Mech Cohes-Frict Mater 1999;4:361–89. prEN 10138-2. 2000. Prestressing steels – Part 2: Wire. European Committee for Standardization. Tork B. Study of the bond-splitting combined mechanisms in reinforced and prestressed concrete elements. Ph.D. Thesis, Univ. Politécnica Madrid; 1999 [in Spanish]. Hillerborg A, Modeer M, Petersson P. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 1976;6:773–82. Galvez JC, Tok B, Planas J. Concrete splitting during release of prestressing force in indented wires. In: Balázs G, Bartos P, Cairns J, Borosnyói A, editors. Bond in concrete – from research to standards; 2002. p. 95–102. Bazˇant ZP, Planas J. Fracture and size effect in concrete and other quasibrittle materials. New York: CRC Press; 1998. ASTM C/C498-04e1. Standard test method for splitting tensile strength of cylindrical concrete specimens. ASTM International; 2004. RILEM 50-FMC Committee Fracture Mechanics of Concrete. Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams. Mater Constr 1986; 18: p. 285–90. Carol I, Prat P, López M. Normal/shear cracking model: application to discrete crack analysis. J Eng Mech ASCE 1997;123:765–73. Gálvez JC, Cervenka J, Saouma V, Cendón DA. A discrete crack approach to normal/shear cracking of concrete. Cem Concr Res 2002;32:1567–85. Gálvez JC, Cendon DA, Planas J. Influence of shear parameters on mixed-mode fracture of concrete. Int J Fract 2002;118:163–89. Cervenka J. Discrete crack modelling in concrete structures. Ph.D. Thesis, University of Colorado; 1994. UNE 36094. Steel wires and strands for prestressing concrete. AENOR; 1997.
Contents lists available at ScienceDirect
Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Splitting failure of precast prestressed concrete during the release of the prestressing force J.C. Gálvez a,*, J.M. Benítez b, B. Tork c, M.J. Casati d, D.A. Cendón e a
Departamento de Ingeniería Civil: Construcción, E.T.S. de Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, 28040 Madrid, Spain Departamento de Vehículos Aeroespaciales, E.T.S. de Ingenieros Aeronáuticos, Universidad Politécnica de Madrid, 28040 Madrid, Spain Ain Shams University, El Cairo, Egypt d Departamento de Vehículos Aeroespaciales, E.U. Ingeniería Técnica Aeronáutica, Universidad Politécnica de Madrid, 28040 Madrid, Spain e Departamento de Ciencia de Materiales, Universidad Politécnica de Madrid, E.T.S. Ingenieros de Caminos, Canales y Puertos, C/Profesor Aranguren s/n, 28040 Madrid, Spain b c
a r t i c l e
i n f o
Article history: Received 23 February 2009 Accepted 10 April 2009 Available online 8 May 2009 Keywords: Concrete Prestressed concrete Precast concrete Bond-splitting Fracture
a b s t r a c t Bond between steel and concrete is fundamental for the transmission of stresses between both materials in precast prestressed concrete. Indented wires are used to improve the bond in these structural elements. The radial component of the prestressing force, increased by Poisson’s effect, may split the surrounding concrete, decreasing the wire confinement and diminishing the bonding. This work presents a testing procedure to obtain the bond–slip curves, between steel and concrete, during the releasing of the prestressing force. The experimental procedure allows study of the splitting failure of the concrete, induced by the action of the indented wire. The influence of the distance between wires, the thickness of concrete cover and the depth of the wire indentations on the bond and splitting are examined. Specimens with three depths of the wire indentations and three thicknesses of concrete cover were tested. Moreover, a numerical procedure is presented for modelling the bond–slip, taking into account the possible failure of concrete by splitting. The numerical procedure accurately reproduces the experimental records and improves knowledge of this complex process. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Precast prestressed concrete elements are widely used for construction in Europe. Nowadays, hollow core slabs and prestressed joists are profusely used for concrete structures in building construction. In both cases, the prestressing force is based on the bond between steel (wire or strand) and concrete. If attention is focussed on single wire, the estimated annual production in Europe of such single wire (4 and 5 mm diameter) for prestressed concrete exceeds 120,000 metric tons [1]. Single wire is annually used to cast more than 45 106 m of prestressed joists and about 15 106 m2 of hollow core slabs in Europe. The case of strands is more difficult to quantify, given that it is used for prestressed and post-tensioned concrete elements. Bond between prestressed steel and concrete is essential for the success of the prestressing system. Nevertheless, bond is a complex problem and depends on many parameters. Due to its significance for practical design, it has been investigated by technical committees, e.g. see Report FIB [2], with it being an attractive challenge for many researchers, as the long list of papers on the topic corroborates. * Corresponding author. E-mail addresses: [email protected] (J.C. Gálvez), [email protected] (J.M. Benítez), [email protected] (B. Tork), mariajesus. [email protected] (M.J. Casati), [email protected] (D.A. Cendón). 1350-6307/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2009.04.023
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
2619
Precast prestressed concrete structural elements are usually made as follows. The steel (strands or single wires) is prestressed in a large precast prestressing table by means of anchor heads. Such a table is usually larger than 100 m or even 200 m. The concrete is then cast and cured with an accelerated curing process. The prestressing force is released by approaching the anchor heads. Finally, the steel between individual structural elements is cut and the prestressing force transmitted to each one. The mechanisms that contribute to the bond between prestressed steel and surrounding concrete are chemical adhesion, friction and mechanical interlocking between wire indentations and concrete [3]. The geometry of the surface of the wire is extremely important for bond. Fig. 1 shows the indention geometry of the wire, and Table 1 shows the dimensions of the indentations. The interlock mechanism may be explained through an analogous mechanism such as that proposed by Tepfers [4,5] for bars in the reinforced concrete (see Fig. 2). When the prestressing force of steel is released at the end of the element by cutting the wire, it tends to pierce in the concrete and develops tangential stress (bond stress) and radial stresses at the interface between the steel and concrete. Both stresses may be related by means of the a angle (see Fig. 2). Furthermore, Hoyer’s effect is present at the ends, is directly related with Poisson’s effect. In the case of indented wires, the wedging action generated by Poisson’s effect is magnified by the indentations on the surface of the wire, increasing the tension ring on these local zones [6–9]. These aspects are beneficial for bond, though they may be harmful if concrete splits, dropping the confinement and diminishing the bond [6,7,10–13]. Bond behaviour of reinforced concrete has been examined by many researchers, such as Gambarova and Rosati [7], Abrishami and Mitchell [8], Tepfers and Olson [5], Ogura et al. [14], Jendele and Cervenka [15] and Malvar [16], among others. Less attention has been devoted to prestressed concrete elements. In this field, bond stress versus slip curves has been studied: den Uijl [10], Abrishami and Mitchell [17] and Tassios and Bonataki [18], with a certain amount of attention being paid to the splitting action of the radial stresses and loss of confinement induced by longitudinal cracking. Several researchers have adopted an analogy between the splitting action of the reinforcement in bond and the pressure of a liquid in a pipe or sleeve [19]. These are two dimensional plane strain models, which are focussed on concrete fracture with no relationship between steel sliding and radial stresses being included, with even Poisson’s effect being omitted [20,21]. This paper presents a test procedure to evaluate the bond-splitting when the prestressing force is transmitted by releasing the steel (wire or strand) in precast prestressed elements. The influence of the distance between wires, the thickness of the concrete cover and the depth of the wire indentations in the bond-splitting is studied. Twenty-seven specimens, combining three thicknesses of concrete cover and three depths of the wire indentations have been tested. This work also includes a numerical procedure to simulate the bond and the possible coupled splitting process. The bond model is conceptually based on the proposal of Cox and Hermann [22–24] for ribbed steel bars by means of an interface with a non-associative plasticity model. Our model is performed for prestressed concrete and only includes parameters that can be experimentally measured. The bond model is also a non-associative plasticity model incorporated in an interface finite element. This bond model is coupled with a fracture model for concrete to take into account the splitting of the concrete. The experimental aforementioned results are used to check the model. This work is not intended to be either a substituting or modelling of transmission length, or one that and evaluates bond strength by standardised testing methods. With this paper the authors seek to emphasise that with this new approach, it is possible to analyse the influence of the parameters affecting bond in the splitting failure of the concrete. It is acknowledged that further work should be carried out to extend this modelling to the majority of the prestressed concrete structural elements. Whereas the following section presents the experimental program, Section 3 examines the experimental results. Numerical procedure is then presented in Section 4, with the model validation being presented in Section 5. Discussion is presented in Section 6. Finally, the conclusions obtained from the model and the experiments are presented in Section 7. 2. Experimental programme Fig. 3 shows two engineering examples of the bond splitting failure in precast prestressed concrete joists. In both cases, the splitting failure, caused by the bond, was detected during the release of the prestressing force. It is worth noting that, due to Hoyer’s effect, the splitting effect is more marked in the edges of the joists (Fig. 3b).
Fig. 1. Geometry of the indentations of the wire [36].
2620
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
Table 1 Specified indentation dimensions, according to UNE 36094 [36]. Nominal wire diameter (mm)
Dimensions of indentations Depth a (mm 102) Type 1
3 4 5 6 P7
Length l (mm)
Spacing p (mm)
3.5 ± 0.5
5.5 ± 0.5
5.0 ± 0.5
8.0 ± 0.5
Type 2 2–6
3–7 4–8 5–10 6–12
5–9 6–10 8–13 10–20
Fig. 2. Bond behaviour of deformed bars, radial component of the bond stresses balanced by tensile stresses in the uncracked ring of concrete [4].
The experimental program was performed to analyse in laboratory, by a simplified modelling, the observed coupled bond-splitting process (Fig. 3a). Fig. 4 shows a sketch of the joist with the geometry and dimensions adopted for the specimens in the experimental analysis. For the sake of simplicity, specimens with only one wire and a rectangular shape cross section were performed. 2.1. Materials and specimens The specimens were manufactured with concrete, composed of Portland cement, siliceous sand and siliceous crushed coarse aggregates of 6 mm maximum size. Table 2 shows the composition of the mixture. Table 3 shows the mechanical properties of the concrete at the test age. EN-10138 Y 1770 C 4 I [25] wires were selected. Table 3 shows the mechanical properties of the wires, being the nominal diameter 4 mm. Wire with three indentation depths (shallow, medium and deep) were used. Table 4 shows the wire indentation depths. Twenty-seven prismatic specimens were cast with a wire embedded longitudinally in the specimen. The dimensions of the prismatic specimens were 400 mm in length, 60 mm in width, and with three different concrete covers. Fig. 5 shows the geometry and dimensions of the specimens. The cross section of the specimens had the shape of a slender rectangle to obtain only two splitting cracks, perpendicular to the longer sides of the rectangle. Table 5 shows the specimen nomenclature and the concrete cover. The thinnest cover was adopted in accordance with the smallest distance between wires in a common prestressed joist or hollow core slab. Before casting the concrete, the wires were tensioned to 17 kN in stiff vertical steel frames. The wire was fixed to the steel frame by wedge anchorages; the lower one was directly supported by the bottom plate, and the upper one fixed to a displaceable externally threaded piece (Fig. 6a). The wire was stressed in a servo-controlled testing machine. The prestressing force was applied monotonically up to the desired value. To keep the prestressing force in the wire, a U-shaped washer was placed between the upper plate and the displaceable anchorage, the nut of the threaded piece being tightened against the Ushaped washer. Then the testing machine was unloaded, transmitting the force to the prestressing frame, and this was then removed from the testing machine. The final prestressing stress of the wire was measured and controlled by a displacement extensometer placed on the central part of the wire. Fig. 6a also shows the safety device added as a protection against accidental snapping of the wire. Once the wires were prestressed, the specimens were cast horizontally in one layer in the ground steel moulds, jointed to the prestressing frames by screws. This screw device guaranteed the alignment of the wire with the longitudinal axis of the specimen and the horizontality during casting and vibrating. The concrete was compacted in the moulds on a vibrating table. The specimens were left in the moulds for 24 h, covered with saturated sacking at room temperature, and then covered with three coats of waterproof paint to maintain humidity during the curing process. The specimens were tested 28 days after casting. For detailed information on specimen preparation, see [26].
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
2621
Fig. 3. Two examples of splitting failure in precast prestressed joists.
2.2. Experimental procedure The tests were performed in two stages: first, the prestressing force was transmitted from the frame to the testing machine, and then the prestressing force from the testing machine to the concrete prism by the controlled release of the prestressing force of the wire. Fig. 7 shows a sketch of the prestressing frame, with the specimen in the testing machine. At the beginning of the first stage, the wire was stressed, the prestressing frame was compressed, and the concrete prism was unstressed. The prestressing frame was coupled to the testing frame of the machine. A tensile load was applied by displacing the machine actuator downward at a rate of 0.1 mm/min till the prestressing force (17 kN) was reached. When the force was completely transmitted to the testing machine, the prestressing frame and the concrete prism were unloaded. Finally, the nut of the threaded moving anchor was loosened and the U-shaped separator was taken off. In the second stage, the prestressing force of the wire was transferred, under control, to the concrete prism, by moving the actuator upward at a rate of 0.3 mm/min. The gradual release of the wire transferred the force to the concrete prism. The test ended when the free ends of the wire were completely unloaded. During the tests the following parameters were recorded: Released load supplied by the testing machine. Displacement of the actuator of the testing machine. Longitudinal shortening of the concrete prism, measured with a gage length of 387.5 mm. Measurements were taken on opposite faces and the mean value was recorded. Wire–concrete slip on the upper and lower faces of the prismatic specimens.
2622
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
Fig. 4. Sketch of a joist with a longitudinal splitting crack. The part selected for the study is indicated by a different weave.
Table 2 Composition of the concrete mixture. Component
Quantity (kg/m3)
Mix proportion by weight
Cement Water Sand Coarse aggregate
450 225 900 720
1.00 0.50 2.00 1.60
Table 3 Mechanical properties of the materials. Concrete
Steel wire
E = 38 GPa fc = 57 MPa ft = 3.0 MPa GF = 100 N/m
E = 226 GPa r0.2 = 1755 MPa ru = 1935 MPa eu = 5.25%
Table 4 Depth of the wire indentation. Denomination
Indentation depth (mm)
Shallow Medium Deep
0.01–0.02 0.04–0.06 0.1–0.11
Crack opening displacement of the longitudinal cracks. Measurements were taken on opposite faces, those with the thinnest cover. 2.3. Testing equipment The prestressing load was applied by a servocontrolled testing machine, and was measured with a 25 kN load cell with ±0.5% error at full scale. Extensometers, of ±2.5 mm nominal travel and ±0.15% error at full scale displacement, were used to measure the shortening of the specimen, the longitudinal crack opening displacement, and wire–concrete slip on the upper and lower faces of the prismatic specimen (see Fig. 8 for details of the placement of the extensometers). 3. Experimental results Fig. 9a–c shows the experimental records of released load versus longitudinal shortening of the concrete prism, for specimens with shallow, medium and deep wire indentations, and concrete covers of 5, 9 and 13 mm. The recorded curves of the specimens with the thinnest concrete cover show a point where the slope of the curves markedly changes. Behind this point the longitudinal shortening of the concrete prism diminishes while the testing machine releases the tension of the wire. This special point marks the beginning of the longitudinal splitting of the concrete prism. The splitting cracks reduce the friction
2623
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
(the wire coincides with the longitudinal axis of the concrete prism) Fig. 5. Geometry and dimensions of the specimens.
Table 5 Dimensions of the tested specimens (see Fig. 5). Specimen
Thickness of specimen (mm)
Concrete cover (mm)
C1 C2 C3
14 22 30
5 9 13
Fig. 6. (a) Detail of displaceable anchorage of the wire for prestressing and (b) horizontal steel mould attached to the frame with the prestressed wire.
2624
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
Fig. 7. Sketch of the test setup.
between the steel and concrete due to the loss of the wire confinement, and consequently induce a release of part of the compressive force in concrete, which diminishes the longitudinal shortening of the concrete prism, as seen in the experimental records. Specimens with 9 and 13 mm thickness concrete cover do not show this behaviour. Fig. 10 compares the experimental records of released load versus longitudinal shortening of the specimens with the thinnest concrete cover and shallow, medium and deep wire indentations. The deepest indentation gave the smallest load of concrete splitting, which suggests that deeper indentation leads to a more intensive tension ring in the concrete, owing to the mechanical interlock between the steel indentations and the concrete. The average splitting loads were 15, 13, 12.5 kN for specimens with wires of shallow, medium and deep indentations, respectively. Moreover, before concrete splits, the deepest the indentation the greatest shortening of the concrete prism. Fig. 11 shows the experimental records of the released load versus crack opening displacement (COD) of the splitting crack, measured on the upper face of the specimen (extensometer 3 in Fig. 8). Each curve shows a break point corresponding to the opening of the crack, shown by a fast rise of the extensometer measurement. The break point load coincides with that in Figs. 9a–c and 10, and confirms the interpretation of those figures. Fig. 12 shows the experimental records of released load versus penetration of the wire into concrete, measured on the upper face of the specimens (extensometer 2 in Fig. 8), for specimens with deep wire indentation, and covers of 5, 9 and 13 mm. Specimens with a concrete cover of 5 mm show a marked change in the slope of the curve. This point corresponds to the start of splitting and complements the above comments for these specimens. The splitting leads to a loss of confinement of the wire, increasing the slip between concrete and steel, with a greater penetration of the wire into concrete. Similar behaviour was observed in the specimens with shallow and medium depth wire indentation. Although specimens with shallow indentation showed a smoother change in the curves than ones with medium and deep indentation. Experimental curves of load versus penetration of the wire in the concrete, measured on the bottom face of the specimens, were practically equal to those of the upper face.
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
2625
Fig. 8. Extensometers and measure devices for: (1) longitudinal shortening, (2) penetration of the wire in the upper face of the concrete prism and (3) COD of the splitting cracks.
4. Numerical procedure 4.1. Problem posing Two processes have to be modelled: (1) the possible splitting of the concrete induced by the radial pressure of the wire and (2) the bond in the interface between concrete and steel. Both processes are related. In this work the model of the splitting concrete failure is based on the cohesive crack approach [27], and that of the bond on a plasticity formulation [26,28]. Fig. 13 shows a sketch of both processes to be modelled. 4.2. The cohesive crack model The cohesive crack model is generally accepted as a realistic simplification of the fracture of quasi-brittle materials. Such a model was proposed by Hillerborg et al. [27] in the late seventies, and has been successful in the analysis of the fracture of concrete and concrete-like materials since its proposal. The softening function, r = f(w), is the main ingredient of the cohesive crack model. This function, a material property, relates the stress r acting across the crack faces to the corresponding crack opening w (see Fig. 14). In mode I opening, the stress transferred, r, is normal to the crack faces. A detailed study of this model has been published by Bazant and Planas [29].
2626
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
a
16
Released load (kN)
14 12 10 8 6 Shallow indentation 4 2 0 -0.2
Cover=5 mm Cover=9 mm Cover=13mm
-0.15
-0.1
-0.05
0
Longitudinal shortening (mm)
b
16
Released load (kN)
14 12 10 8 6 Medium indentation 4 2 0 -0.2
Cover=5 mm Cover=9 mm Cover=13mm
-0.15
-0.1
-0.05
0
Longitudinal shortening (mm)
c
16
Released load (kN)
14 12 10 8 6 Deep indentation 4 2 0 -0.2
Cover=5 mm Cover=9 mm Cover=13mm
-0.15
-0.1
-0.05
0
Longitudinal shortening (mm) Fig. 9. Released load versus longitudinal shortening of the concrete prism in specimens with 5, 9 and 13 mm of thickness concrete cover and the wire indentation: (a) shallow, (b) medium and (c) deep.
Two properties of the softening curve are most important: the tensile strength, ft, and the cohesive fracture energy, GF. The tensile strength is the stress at which the crack is created and starts to open (f(0) = ft). The cohesive fracture energy, GF, also called specific fracture energy, is the external energy supply required to create a full break unit surface area of a cohesive crack, and coincides with the area under the softening function. The tensile strength and the specific fracture energy are material properties and may be experimentally measured in accordance with ASTM C 496 [30] and RILEM 50-FMC [31], respectively.
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
2627
16
Released load (kN)
14 12 10 8 6 Cover=5 mm
4
Shallow indentation Medium indentation Deep indentation
2 0 -0.2
-0.15 -0.1 -0.05 Longitudinal shortening (mm)
0
Fig. 10. Released load versus longitudinal shortening of the concrete prism in specimens with 5 mm of thickness concrete cover and three depths of the wire indentation.
16
Released load (kN)
14 12 10 8 6
Cover = 5 mm
4
Shallow indentation Medium indentation Deep indentation
2 0 0
0.05
0.1
0.15
0.2
0.25
COD (mm) Fig. 11. Released load versus COD of the splitting crack of the concrete prism in specimens with 5 mm of thickness concrete cover and three depths of the wire indentation.
4.3. The bond model The interface between concrete and steel is able to transmit normal and tangential stresses and shows dilatancy. The barscale approach, according Cox’s terminology [23,26], is adopted, then the wire–concrete interface is idealized in terms of two simplifications: (1) uniform distribution of the bond stress for a unit surface element (Fig. 15a), and (2) idealised deformation of the concrete in the bond zone (Fig. 15b). 4.3.1. Yield surface for bonding The relative displacement between the wire and concrete at the interface is vectorial in nature. We denote it as u, with normal and shear components denoted as un and ut, i.e.,
n þ ut~ u ¼ un~ t
ð1Þ
where ~ n and ~ t are the unit vectors, respectively, normal and tangential to the surface of the wire. Likewise, the stress transferred on the interface between the concrete and the wire is also vectorial, and is characterised by the vector t acting on the interface; with normal and tangential components denoted as r and s, i.e.,
n þ s~ t ¼ r~ t
ð2Þ
In this work, elastoplastic formulation is adopted, in which the interface displacement is split into its elastic and inelastic parts
u ¼ u e þ ui ;
u_ ¼ u_ e þ u_ i
ð3Þ
2628
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
16
Released load (kN)
14 12 10 8 6 Deep indentation
4
Cover=5 mm Cover=9 mm Cover=13mm
2 0 0
0.2
0.4 0.6 0.8 1 Wire penetration (mm)
1.2
1.4
Fig. 12. Released load versus wire penetration into concrete prism in specimens with 5, 9 and 13 mm of thickness concrete cover and deep depth of the wire indentation.
Fig. 13. Numerical problem processes: (a) longitudinal cracking of concrete (splitting) and (b) bond–slip behaviour of the interface between wire and concrete.
Fig. 14. Cohesive crack, softening function and notation for mode I fracture of quasi-brittle materials.
For the inelastic behaviour, it is assumed that the inelastic displacement can progress when the yield surface F(t) = 0 is reached. The following hyperbolic expression [32] has been assumed by the authors [33,34]:
FðtÞ ¼ s2 tan /f ðft rÞ 2c tan /f ðft þ rÞ
ð4Þ
where /f is the friction angle between the wire and concrete, c the cohesion (bond strength without interface normal stress), and ft the tensile strength in the normal direction to the interface. These values are instantaneous values and depend on loading history through the effective inelastic crack displacement uieff, defined by the conditions
0:5 u_ ieff ¼ ku_ i k ¼ u_ 2n þ u_ 2t Z uieff ¼ u_ ieff dt
ð5Þ ð6Þ
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
2629
Fig. 15. Sketch of the idealised interface between wire and concrete for a unit surface element: (a) uniform distribution of the bond stress and (b) deformation of the concrete in the bond zone.
In this work it is assumed that the friction angle /f is constant, while the instantaneous tensile strength ft and cohesion c depend on uieff bilinearly as depicted in Fig. 16. Following the analogy with the cohesive model, the area under the softening curve for ft may be interpreted as the specific fracture energy GIF for mode I, and the area under the softening curve for the cohesion c (also has dimensions of energy per unit area) as the mode IIa (shear under normal confinement) specific fracture energy GIIa F [35]. For the particular case of bond, this one may be measured by means of a pull-out test under high confinement. The following values for xc and xs in the softening curves (see Fig. 6) were adopted.
xc ¼ xr ¼
2GIIa F ðs1c þ c0 Þx1c s1c 2GIF
ðs1r þ ft0 Þx1r s1r
ð7Þ ð8Þ
Fig. 17 shows the plasticity surface for bonding in the interface, and its evolution with the parameter uieff. Note that for each state of damage, the hyperbolic plasticity surface has two branches and only the branch extending towards negative values of r is physically acceptable. Furthermore, it should also be noted that for fully damaged interface (complete loss of tensile strength and cohesion), the plasticity surface degenerates into a Coulomb friction surface with friction coefficient l = tan /f. 4.3.2. Flow rule and dilatancy Due to the dilatant behaviour of the wire-steel interface, non-associative plasticity has been adopted. Then, the evolution of the inelastic displacements in the interface zone is specified by means of the flow rule, given by
u_ i ¼ k_
@QðtÞ _ ¼ kb @t
ð9Þ
Fig. 16. Softening curves for the strength of the interface between wire and concrete: (a) normal direction (tensile strength) and (b) tangential direction (bond strength).
2630
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
Initial plasticity surface (u
ieff
= 0)
τ Intermediate plasticity surface (0 < u
φf
ieff
Final plasticity surface ieff
( u > wσ ) σ
Fig. 17. Plasticity surface and evolution.
where Q(t) is the plastic potential, b is the normal to the plastic potential surface and k_ is a non-negative plastic multiplier. The dilatancy angle is also defined
tan /d ¼
u_ in u_ it
ð10Þ
where /d is the dilatancy angle, u_ in and u_ it are the incremental inelastic displacements in normal and tangential directions to the interface, respectively. /d coincides with the angle formed by the potential surface and the negative part of the r axis (see Fig. 18). The dilatancy is also assumed to depend on the damage level through uieff. Following Ref. [35,36], a linear curve has been adopted:
8 < / 1 uieff 8uieff < ucd d0 ucd /d ¼ : 0 8uieff P ucd
ð11Þ
where /d0 is the initial value of the dilatancy angle and ucd is the critical inelastic crack displacement after which the interface ceases to exhibit the dilatancy effect.
φ
d
b
B φ
d
A φ
d
b
O
φ
d
A' φ
d
φ
d
b
B'
Fig. 18. Return direction to the plasticity surface of the inelastic corrector.
2631
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
When traction stress r is predominant over the tangential stress s, the direction normal to the plastic potential (Q = cte) is not defined. Consequently, in this case the return direction to the origin of stresses is adopted, which divides the stresses space into two parts, as is shown in Fig. 18. It should be pointed out that this is a non-associative plasticity approach, where normal directions to cracking surface and plastic potential are different. 4.3.3. Incorporation into a finite element code An interface element was developed to incorporate the bond model in the finite element code ABAQUSÓ. The cohesive model for simulation of the radial cracks was included by means of non-linear springs. Details regarding the finite element implementation are found in [26]. An arc length algorithm was adopted for the computational procedure and no special difficulties were found in achieving a convergent solution. Fig. 19 shows a sketch of the finite element mesh. To simplify the computations, the bulk behaviour of the material outside of the crack and the interface was assumed to be linear-elastic and isotropic, although this approximation could have been relaxed if necessary. 5. Model validation The presented numerical procedure is used to reproduce the experimental results. Table 3 shows the mechanical properties of the materials, and Table 6 the parameters adopted for the interface for the three depths of the indentations. These parameters were not experimentally measured, but estimated. Fig. 20 shows the experimental records and the numerical prediction of released load versus longitudinal shortening of the concrete prism, for specimens with shallow, medium and deep wire indentations, and the minimum concrete cover (5 mm). Fig. 21 compares the experimental records and the numerical prediction of released load versus COD of the possible splitting crack on the concrete prism, for specimens with shallow, medium and deep wire indentations, and the minimum concrete cover (5 mm). Fig. 22 shows the experimental records and the numerical prediction of released load versus wire penetration on the edge of a concrete prism, for specimens with medium wire indentations, and the minimum concrete cover (5 mm). In all cases the numerical prediction properly predicts the bond behaviour and the coupled splitting failure. The prediction fit quite well the entire behaviour of the experimental records. 6. Discussion There is agreement that, in reinforced concrete with ribbed bars, the splitting failure occurs when the concrete cover is lower than three times bar diameter [12]. This agreement is based on experimental pull-out results with ribbed bars, and analytical and numerical studies of the cross section of the element with an internal pressure on the bar hollow. However, there is not a detailed study for the bond-splitting on precast prestressed concrete, taking into account the bond parameters and their role in the splitting action. The proposed experimental procedure allows for study of the splitting failure and the bond in the interface steel-concrete as combined processes. This work concerns itself only with bond-splitting failure and is not intended to be a substitute of any method of bond strength or transmission length in prestressed concrete. The experimental results of the specimens with the thinnest concrete cover have shown that the depth of the indentations is crucial as regards the possible splitting failure. The smallest thickness of the concrete cover, 5 mm (1.25 times wire
Fig. 19. Sketch of the finite element mesh used for the numerical analysis.
Table 6 Parameters of the interface adopted for numerical modelling. Denomination
smax (MPa)
C0 (MPa)
/d (°)
/f (°)
Shallow Medium Deep
10 10 10
65.3 64.2 63.8
0.3 4 16
40 42 43
2632
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
a
16
Released load (kN)
14 12 10 8 6 4 2 0 -0.2
Shallow indentation Cover = 5mm Experimental Numerical
-0.15
-0.1
-0.05
0
Longitudinal shortening (mm)
b
16
Released load (kN)
14 12 10 8 6 4 2 0 -0.2
c
Medium indentation Cover = 5 mm Experimental Numerical
-0.15 -0.1 -0.05 Longitudinal shortening (mm)
0
16
Released load (kN)
14 12 10 8 6 4 2 0 -0.2
Deep indentation Cover = 5 mm Experimental Numerical
-0.15 -0.1 -0.05 Longitudinal shortening (mm)
0
Fig. 20. Experimental records and numerical prediction of released load versus longitudinal shortening of the concrete prism in specimens with 5 mm of thickness concrete cover and wire indentation: (a) shallow, (b) medium and (c) deep.
diameter), is apparently much thinner, but is based on the real dimensions used by industry for some of these precast prestressed joists (see Fig. 3). The comparison of the experimental records of the released load versus specimen shortening, COD and wire penetration has shown that the deepest indentation the highest splitting action of the wire. The numerical modelling accurately reproduces the experimental records. The fracture cohesive model only includes parameters that have been measured by standardised methods. The bond model includes parameters that may to be experimentally measured. In this work, some parameters for bond modelling (see Table 6) have been estimated and not experimentally measured. These adopted values look according with the experimental results, and may be obtained by an inverse analysis, out the scope of this work.
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
2633
Shallow indentation
16 Medium indentation
Released load (kN)
14 12 Deep indentation
10 8 6
Cover = 5 mm
4
Numerical Experimental
2 0 0
0.04
0.08 COD (mm)
0.12
Fig. 21. Experimental records and numerical prediction of released load versus COD of the splitting crack of the concrete prism in specimens with 5 mm of thickness concrete cover and three depths of the wire indentation.
16
Released load (kN)
14 12 10 8 6
Medium indentation Cover = 5 mm
4
Experimental Numerical
2 0 0
0.2
0.4 0.6 0.8 Wire penetration (mm) Upper face
1
1.2
Fig. 22. Experimental records and numerical prediction of released load versus wire penetration into concrete prism in specimens with 5 mm of thickness concrete cover and medium depth of the wire indentation.
7. Conclusions and final comments A testing procedure to study the splitting failure of precast prestressed concrete elements during the release of the prestressing force has been proposed. The procedure allows studying the parameters affecting the bonding in the splitting failure of the concrete. Twenty-seven specimens, combining three thicknesses of concrete cover and three depths of the wire indentations, have been tested. The following remarks may be made from the experimental work: Splitting failure has been observed on the specimens with the thinnest concrete cover. This failure was observed by means of a change in the slope of the released load versus longitudinal shortening of the specimen recorded curves. The longitudinal shortening of the specimen decreased after splitting failure, indicating the loss of wire confinement. The critical released load that showed the splitting failure diminished when the depth of the wire indentations increased. The COD measurement has made easier to detect and follow the splitting failure. Moreover, it has been experimentally observed that the deepest depth wire indentation the earliest (load released) and largest COD in the longitudinal cracking failure. The deepest depth wire indentation the best bond between wire and concrete, until the splitting failure. This better behaviour is shown by a lower sliding of the wire at the end of the specimen and by a larger longitudinal shortening of the specimen for equal released load in the recorded experimental curves. It is worth noting that the deeper wire indentation leads to a higher splitting stresses and to a larger COD in the specimens. A numerical procedure for modelling the bond–slip at the interface of the wire and concrete has been proposed. The procedure takes into account the possible failure of concrete by the splitting action of the wire. The bond modelling is based on a
2634
J.C. Gálvez et al. / Engineering Failure Analysis 16 (2009) 2618–2634
non-associative plasticity approach, and the splitting concrete modelling is based on the cohesive crack approach for quasibrittle materials. The bond model was incorporated into an interface finite element. The cohesive crack model for the simulation of the radial cracks was included by means of non-linear springs. Both models were incorporated in the finite element code ABAQUSÓ. The numerical procedure has been contrasted with the above experimental results, and accurately reproduces the experimental records. The experimental work and the numerical procedure emphasise that the cohesive crack models, in combination with the bond modelling, are promising tools in the simulation of the splitting failure of the precast prestressed concrete structural elements. Further more work must to be carried out to extend this modelling to full-scale structural elements. Acknowledgements The authors gratefully acknowledge the financial support for the research provided by the Spanish Ministerio de Educación y Ciencia under Grants BIA-2005-09250-CO3-02 and BIA-2008-03523 and by the Ministerio de Fomento under Grants MFOM-2004/9 and MFOM-01/07. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]