INTRODUCTION

Ultra-high performance concrete (UHPC) is a material with outstanding durability, a high-compressive strength, a sustained post-cracking tensile strength and notable self-compacting capacity. UHPC elements exhibit compressive and tensile properties higher of those from conventional and fiber-reinforced concretes. Currently, UHPC precast construction is regarded as an interesting alternative to be considered in civil engineering projects to take advantage of its exceptional material properties. Discovering a practical and reliable method of connecting ultra-high performance concrete precast elements with fasteners has been recognized as a critical factor in future construction projects (Maya et al. [1]).

Analytical, numerical and experimental methods have been used to investigate dowelled joints in anisotropic plates [2-11].

Echavarría and Salenikovich [12], presented an analytical model to calculate the stresses around a hole in a bolted joint and to predict failures in timber bolted connections. Timber bolted joints loaded parallel to grain were tested. Experimental results agreed well with theoretical predictions.

In Hammoud and Naaman [13], experimental work was carried out in order to study the behavior of ferrocement bolted shear joints. The type, number, and orientation of mesh layers, along with end distance and hole diameter were studied. Equations were developed to predict the net tension, cleavage, and bearing failure loads.

Maya et al. [1], have studied UHPC precast elements. The behavior of precast elements connected using short reinforcement splice lengths was analyzed. This configuration reduced the in-situ work.

Mansur et al. [14], have described an experimental investigation on ferrocement bolted joints. Laboratory tests have shown that the mode of failure of a moment joint depends on whether the applied moment is in the opening or closing mode.

Kwon et al. [15], have experimentally investigated the joint strength of bolted UHPC elements. The width, the thickness, and the distance from the center of the hole to the edge of the joint were considered. Results showed that the joint strength does not increase when the size and thickness of the joint augment.

Unfortunately, the research in bolted UHPC joints is so far insufficient (Camacho et al [16]). This lack of studies has delayed the development of an analytical design criteria.

The objective of this paper is to experimentally validate, in an UHPC joint, the analytical method, presented initially by Echavarría [17] capable of determining the stress distribution around a pin-loaded hole to predict brittle failure modes.

1. DEVELOPMENT OF BASIC EQUATIONS

The stress concentrations in an elastically orthotropic plate are examined for a joint loaded frictionless along the edge of the hole by an infinitely rigid pin (see Fig. 1 and Fig. 2). The loading force F is of magnitude pdt. Where d is the diameter of the hole, p is the bearing stress, and t is the unit thickness of the plate.

Figure 1
Joint geometry

Figure 2
Joint under a sinusoidal distribution of load.

In general, the boundary conditions are used to calculate the stress functions. The stresses for the joint under consideration are summarized below.

The stresses at (x = 0, y = d/2) are calculated in this way:

Stress concentrations calculated with this analytical method are identical with those of the fundamental method found in De Jong [18] and Bickley [19]. The accuracy of De Jong's solution falls off in view of the excessive number of calculations. On the other hand, Bickley's solution considers only infinite isotropic plates.

2. PREDICTING BRITTLE FAILURES OF A BOLTED ULTRA-HIGH PERFORMANCE CONCRETE JOINT

There are four common failure modes in bolted joints, namely tension, shear-out, bearing, and cleavage. Unfortunately, brittle failures in UHPC bolted joints with inadequate end and edge distances and fastener spacing occur predominantly due to tension. In this study, the width of plate (b/d ratio) and the end distance (e/d ratio) were considered. Table 2 and Table 3 show the stress-concentration factors σ_x,max and σ_y,max (normalized by the average bearing stress p) assuming the elastic properties listed in Table 1 of a UHPC element of a unit thickness t and width b. For clarity, the tensile stresses are reported as positive.

Table 2
Orthogonal stress concentration (σ_x,max/ p )

e/d	Equation (1)
	b = ∞	b = 16d	b = 8d	b = 4d
2	0.43	0.40	0.37	0.30
4	-0.46	-0.49	-0.53	-0.59
8	-0.71	-0.74	-0.77	-0.84
16	-1.02	-1.05	-1.08	-1.14

Table 3
Longitudinal stress concentration (σ_y,max/p )

e/d	Equation (5)
	b = ∞	b = 16d	b = 8d	b = 4d
2, 4, 8, 16	0.81	0.90	1.00	1.19

Table 1
Material properties of UHPC

Ex (GPa)	Ey (GPa)	Gxy (Gpa)	Vyx	Tensile strength (MPa)
45	45	18.75	0.20	10

• This analytical solution indicates that the reduction of the b/d ratio will increase the σ_y,max and, simultaneously, will cause a reduction in the σ_x,max .

• Clearly σ_y,max will increase if b/d decreases. When b/d is small, σ_y,max generates a pure tension failure in UHPC joints because of their low ultimate longitudinal tensile strength.

• σ_x,max will cause splitting in materials with a low ultimate orthogonal tensile strength (i.e. wood joints).

• The highest longitudinal compressive stress (Equation 2) is independent of e/d and b/d and can result in the bearing failure of the joint.

Considering a brittle performance of UHPC in tension and the maximum stress criteria, the joint will break predominantly when the highest longitudinal tensile stress σ_y,max reaches the corresponding strength value.

3. EXPERIMENTAL VERIFICATION

Laboratory tests on ultra-high performance fiber reinforced concrete (UHPFRC) joints with the geometry shown in Fig. 1 were conducted to validate the maximum stress concentrations predicted by the model.

An in-house developed UHPFRC mix made with CEM HE ASTM C1157 type cement, silica fume, and a superplasticizer was used for the experiments. The UHPFRC specimens used straight non-deformed steel fibers that were 13 mm long and had a 0.2 mm diameter. These fibers were included in the mix at a concentration of 2% by volume.

Static load applied in tension using a universal testing machine was used. The displacement-control rate of loading was set to reach failure in 5 to 15 minutes. UHPFRC elements for the joints were cut from 1 by 1-m panels. In addition, the elastic properties of UHPFRC panels were determined using ultrasonic pulse velocity tests. Tensile strength was determined using direct tensile tests (Graybeal et al. [20]). The average properties from 50 replications tested are shown in Table 1.

Forty joints were tested using various geometric combinations as specified in Table 4. The bolt was 9.5 mm (3/8 in.) in diameter and was made of low carbon steel. The t/d ratio was small enough to avoid bending deformation of the bolt and induce failure in the UHPFRC element.

Table 4
Experimental results for UHPC bolted joints

e/d	b	A	B	C	D
2	3d	7	1.0p	12	Tension
6	3d	7	1.2p	21	Tension
2	10d	20	0.7p	15	Tension
6	10d	6	0.4p	32	Tension

Summarization of experimental results and the comparison with the analytical predictions are given in Table 4 and Table 5. The specimens failed mostly in pure tension, as predicted by the analytical model (see Fig. 3).

Table 5
Analytical predictions for UHPC bolted joints

where,

e/d	b	E Equation (5)	F
2	3d	1.4p	Tension
6	3d	1.4p	Tension
2	10d	1.0p	Tension
6	10d	1.0p	Tension

Figure 3
Net tension failure.

where A Number of replications B Experimental longitudinal stress concentration C Coefficient of variation (%) D Prevalent failure mode where, E Predicted longitudinal stress concentration F Predicted failure mode

4. CONCLUSIONS

The prediction of stress concentrations in fastened precast concrete plates is a common issue. The focus of this work is to validate an effective analytical method to predict brittle failure modes for different geometries and anisotropic conditions of an UHPC joint. The proposed analytical equations allow estimate the stress concentrations using the elastic properties of the anisotropic material. These predictions correspond reasonably well with those of the analytical solutions found in De Jong [18] and Bickley [19]. Further, laboratory experiments in UHPC joints agree well with the analytical predictions. Most importantly, it is the first time that this analytical model is applied to predict brittle failures of UHPC joints. These results prove that the analytical model originally developed for orthotropic plates can be also applied to UHPC joints. The variability of the experimental dowelled joints results is high. In future projects it would be convenient to increase the number of tests. An efficient and accurate analytical model is certainly an essential precursor to additional experimental studies and numerical models.

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Notes