Based on the theoretical and experimental parts of this research, relevant conclusions on element behaviour and on possibilities for modelling and design of UHPFRC members are presented.
Bending in simply supported UHPFRC beams Testing and modelling
• a test series has been carried out on simply supported UHPFRC beams with thickness varying between 25 and 75 mm and tested in three-point bending;
• an analytical model describing the non linear behaviour of UHPFRC beams has been developed;
• a finite element model (FEM) describing the non linear behaviour of UHPFRC beams has been developed;
• the results of the analytical model are in good agreement with experimental data and with the prediction of the FEM model. Good agreement is also shown with test results from other authors on beams made of tensile strain hardening materials or materials without strain hardening phase;
• based on the comparison between theoretical and experimental results, multi-microcracking, causing tensile strain hardening with low stress increase, can be well modelled using pseudo-plastic material behaviour in tension for the thickness range of tested elements;
• explicit analytical expressions have been developed to describe the non-linear force-deformation response of a beam exhibiting multi-microcracking. Explicit expressions have also been developed to predict progressive reduction of the unloading-reloading stiffness caused by damage during microcracking;
• the propagation of a macrocrack, characterised by stress-crack opening relationship, can be simulated using the hypothesis of the fictitious crack model;
• a simplified formulation of the model is proposed to describe the bending behaviour in presence of macrocrack opening.
Behaviour of UHPFRC beams
According to the result of the analytical model and in agreement with experimental evidence, it is concluded that:
• the pre-peak behaviour and bending strength are mainly governed by multi-microcracking;
for typical material properties, with tensile deformations ranging up to 2.5 ‰, the equivalent bending stress achieved due to strain hardening is approximately 2.4 times the tensile strength; this contribution is size-independent as long as sufficient compressive strength is provided and compression softening is avoided, as is the case of rectangular members in pure bending.
• the propagation of the macrocrack, characterised by tensile softening, provides a minor additional contribution to bending strength (up to roughly 10%), which is a size dependent value; bending strengths are attained for small macrocrack openings (smaller than 0.2 mm for the tested beams);
• in the case of thin beams, propagation of the macrocrack plays an important role in providing ductility at bending failure as well as post-peak toughness;
• from experimental observations, it must also be pointed out that more than one macrocrack can develop prior to peak-force, even though a gradient of moment existed;
• an important portion of the initial elastic structural stiffness of a beam is maintained for a load range that covers service conditions: for up to nearly 80% of the ultimate load, the residual stiffness equals more than 80% of the elastic one for specimens in three-point bending and approximately 60% for specimens under uniformly distributed load.
Influence of material properties and structural size on bending behaviour
The main results of a parametric study carried out using the analytical model can be summarized as follows:
• the size effect on bending strength is much less significant for UHPFRC than for other quasi-brittle materials: experimental results for thin members tested by the author (25 to 75 mm) confirm that size effect on bending strength is practically negligible; similar results are also reported by other authors for thicknesses up to 300 mm and for materials characterised by pronounced pseudo-plastic tensile deformations;
• the size effect on ductility is evident, even in a small range of variation of element thicknesses.
• even if the pseudo-plastic tensile phase is not pronounced, thin elements can develop a behaviour similar to that of elements with high pseudo-plastic tensile deformations, owing to the low stress decrease during tensile softening. As a consequence, if only force-displacement response of thin elements in bending is known, it is unreliable to characterise the material tensile hardening capacity by back analysis;
• in absence of pseudo-plastic tensile deformations, the behaviour of thick elements approaches that of typical quasi-brittle materials, with a pronounced size effect both on bending strength and ductility;
• the influence of the tension softening behaviour on bending strength is limited if the pseudo-plastic phase is pronounced, since most of the bending strength is developed while the concrete is in the pseudo-plastic tensile phase. The tensile softening is however important in providing ductility at failure and post-peak.
Statically indeterminate systems: beams and slabs
Numerical procedures have been developed to simulate the non-linear response of statically indeterminate UHPFRC beams and slabs subjected to bending. Firstly, the case of a clamped beam has been theoretically studied and the following conclusions are drawn:
• load carrying capacity of statically indeterminate members is positively affected by the ductility in bending of thin UHPFRC beams, which allow a significant redistribution of internal forces;
• load carrying capacity of statically indeterminate members is more sensitive to size effect than the bending strength of statically determinate elements. This is due to the fact that the load carrying capacity of a statically indeterminate system depends on the possibility to redistribute internal forces. Redistribution is strictly related to ductility in bending that, as demonstrated for simply supported beams, is controlled by the opening of a macrocrack and is thus much more affected by size effect.
A test series has been carried out on thin UHPFRC square slabs (40-60 mm thickness, 900 mm side length) supported on eight symmetrical points and subjected to central point load: all of the slabs failed in bending. The experimental behaviour was modelled using the same hypotheses for the behaviour of UHPFRC as the hypothesis used for modelling beam elements. The following conclusions are drawn:
• good agreement of experimental and theoretical results on pre-peak behaviour of UHPFRC slabs is shown; this confirms again that multi-microcracking can be well represented as a homogeneous material behaviour using an elastic-pseudo-plastic material law in tension;
• a major part of the load-bearing capacity of the slab is achieved with multi-microcracking behaviour; this is also confirmed by the experiment, during which cracks become visible only for a force level close to maximal force; it seems thus viable to predict the service response of a thin UHPFRC slab using elastic-plastic material tensile law and commercial FEM models;
• when the capacity to develop microcracking is exhausted, deformations start to localise i.e.
discrete macrocracks form and a rigid block mechanism starts to develop. For the given boundary conditions, the non-linear simulation predicts dominant tangential deformations that are in agreement with experimentally observed macrocracks, opening along radii.
Application of the theory of plasticity in design of UHPFRC elements in bending
• A concept of plasticity for thin UHPFRC elements without ordinary reinforcement has been defined based on the bending response in statically determinate systems: the moment-curvature relationship during the first phase of macrocrack opening in thin elements is characterised by an almost constant moment with increasing curvatures. Thus, the moment level at the beginning of macrocrack propagation is assumed as plastic resistant moment, and a rigid-perfectly plastic moment-curvature relationship is defined with a limiting value for the curvature.
Based on this definition, the applicability of theory of plasticity in the design of statically indeterminate UHPFRC members has been thus evaluated:
• according to the results of the non-linear beam analysis, the applicability of the theory of plasticity to UHPFRC elements in bending depends on element size;
• with the proposed definition of plastic resistant moment, the theory of plasticity can be applied to predict failure loads for thin UHPFRC beams up to approximately h <100 mm.
For thicker beams, the deformational capacity of the first developed plastic hinges may not be sufficient to allow the formation of other hinges and the development of a mechanism;
• the ultimate load-bearing capacity of thin UHPFRC slabs without ordinary reinforcement is well assessed based on the theory of plasticity, by applying the yield line method: for the investigated span and thickness range, the comparison with test results validates the proposed definition of plastic resistant moment.
Punching-shear in thin UHPFRC slabs
• A test series has been carried out on small square slabs (300 mm side length and 30-60 mm thickness) clamped on the edges and subjected to central point load on a 20x20 mm2 surface;
• an analytical approach for assessing the punching shear strength of UHPFRC members has been proposed: fibre contribution to shear strength is taken into account as a function of element size and stiffness.
Although current theoretical and experimental knowledge is still insufficient for a general model to be established, the predictions of the proposed approach are in good agreement with the experimental results obtained in the framework of this research and by other authors. On the basis of theoretical and experimental evidence, the following conclusions can be drawn:
• significant shear stresses can be sustained by UHPFRC elements;
• fibre contribution to punching shear strength is a function of the opening of the critical shear crack, which is a function of slab thickness and slab bending stiffness at failure;
• punching shear strength decreases for increasing slab thickness;
• punching shear strength decreases for slabs that develop higher deformations, i.e. long-span slabs or slabs with free boundary conditions; thus, the influence of slab thickness, span and boundary condition should be taken into account to correctly evaluate punching shear strength;
• current design recommendations poorly estimate punching shear strength for element thicknesses lower than 150 mm, and may lead to unsafe predictions in the case of slabs with pronounced deformations;
Structural application of UHPFRC
A state of the art of existing structural application of UHPFRC has been made with specific attention to bridge design. It can be concluded that:
• current structural application of UHPFRC in bridge design often considers classical prestressed concrete shapes, sized with respect to increased material strengths. Principal bending solicitations are sustained in a manner similar as in ordinary concrete, with tensile forces being sustained by prestressing steel. However, savings in concrete quantity in the range of up to three times in comparison to ordinary concrete are achieved and shear reinforcement is usually completely avoided thank to the shear carrying capacity of the fibres;
• the concept of thin walled structures is particularly gainful for application of UHPFRC, since very thin elements can be conceived, providing local resistances, while a sufficient stiffness and global resistance are assured by the prestressed elements, resulting in significantly decreased weight.
The design and optimization of a ribbed deck slab against local and global failure in bending and punching shear is considered as a case study, showing that:
• thin slabs (40-60 mm) restrained by the ribs can safely resist both bending and punching-shear failure introduced by design wheel-load;
• depending on rib spacing and the span between principal girders i.e. slab supports, a ribbed slab with an equivalent height in the range of 110 to 140 mm can be designed for road bridge applications;
• for the design resistance level of prestressed elements coinciding with strand yielding, a high percentage of the initial element stiffness is maintained at service states;
• investigated ribbed deck allows significant material savings (up to three times), that could not be achieved with ordinary concrete.