CALCULATION MODEL FOR STEEL FIBRE REINFORCED CONCRETE PUNCHING ZONES OF BRIDGE SUPERSTRUCTURE AND FOUNDATION SLABS

The usage of steel fibre reinforced concrete in monolithic joins is well known as a good alternative of additional reinforcement because of chaotic distribution of steel fibres in complex stress ant strain state. Unfortunately, the analysis of well known design codes and different models even in punching case without steel fibres shows that there is no common theory in calculating punching shear strength. Existing models of punching shear strength with steel fibres are mainly based on empirical coefficients, or require direct tests, what makes the design of such structures more complicated. Besides, the analysis of elastic and plastic characteristics of steel fibre reinforced concrete is incomplete, because there is no unified, well-grounded theory to evaluate them. The aim of this paper is to present steel fibre reinforced concrete punching shear strength model. Suggested steel fibres reinforced concrete punching shear strength model estimates the main factors, such as concrete strength, longitudinal reinforcement, steel fibres volume, type, geometric and anchoring characteristics, and also plastic strains of steel fibre reinforced concrete. The comparison of suggested model with tests results demonstrates good accuracy of the suggested model for steel fibre reinforced concrete slabs (mean value – 1.12, standard deviation – 0.08 coefficient of variation – 7%).


Introduction
In many cases complicated stress and strain state due to load action in structures of bridges appears. That is the case with joints between flat slabs and columns, columns to foundations joints, connections of columns to piles, etc. The said and other structural bridge members may be subjected to impulse and cycle loads. Various types of stresses in these areas of structures are generated: compression, tension, local compression, shear etc. It is well known that the highest strength of the concrete is in compression and the lowest -in tension. When strengthening of a concrete zone subjected to the action of forces in one direction only is required then one direction reinforcing is applied. But in the case of complicated stress state reinforcing by bars becomes either complicated or even economically unjustifiable. Such state of stress takes place in some bridge structures, e. g. in superstructure decks supported on columns and these -on foundation reinforced concrete (RC) slab ( Fig. 1). For such structures or its elements more effective materials are materials with higher mechanical and deformation properties in all directions of axis. Such material for many RC structures is steel fibre reinforced concrete (SFRC) (Baikovs, Rocēns 2010;Szmigiera 2007;Šalna, Marčiukaitis 2007).
Analytical models and mechanical calculation methods for SFRC were created. Compression and tension strengths and elasticity modulus of SFRC were most widely investigated. They are the main properties that allowed starting using such type of concrete for load bearing RC structures or their parts. In some countries standards and technical recommendations regulating quality and requirements for SFRC products are established: in the USA -ASTM C1018-97 Standard Test Method for Flexural Toughness and First-Crack Strength of Fiber-Reinforced Concrete (Using Beam with Third-Point Loading), in Japan -JSCE- SF4: 1984. Method of Tests for Flexural Strength andFlexural Toughness of Steel Fibre Reinforced Concrete, in the United Kingdom -TR34 Report (1995), General document andAppendix F: Slab Design with Steel Fibres, in Russia -СП 52-104-2006: Сталефибрбетонные Конструкции [SP 52-104-2006: Steel Fibre Reinforced Concrete Structures], etc.
Nevertheless, methods for determination of properties and estimation, values of results according to the said documents differ in many cases; valuation of indices describing intermediate strength as well as rates relating strength and strain are assessed in different way. Generally, stress to strain, especially plastic ones, relation is determined by tests. There is a lack of theoretical and experimental investigations in behaviour of SFRC in elastic plastic stage. It is an obstacle in selecting a suitable model for punching strength analysis of SFRC slabs, e. g. in standards of ASTM C1018-97 and JSCE-SF4: 1984 accomplishment of experiments is proposed for determination of the actual σ-e diagrams.
Investigation in punching strength analysis methods and in design codes of various countries (Šalna et al. 2004) pointed out that the opinion about punching strength analysis varies. According to the all said codes conditional tangential stress (t c ) acting in a conditional critical area is conditionally described and does not reflect actual behaviour of structure at all. Moreover, effect of such important factor as the longitudinal reinforcement is taken in to account not in all codes. Influence of transverse reinforcement is evaluated not in the same way as well. Values of partial safety factors to the strengths of materials and to the loads as well are different.
There is no unified opinion about design for punching strength according to the all codes and there is no unified model for punching strength analysis as well. For description of models various theories were applied -theory of plasticity, of built-up bars, of failure etc. but the unified opinion was not reached. Analysis of models performed by Georgopoulos (1989), Broms (1990), Hallgren (1996), Menétrey (2002), Theodorakopoulos and Swamy (2002) also points out that in all models considered different design diagrams and different criteria are assumed. The principle stresses are related to the concrete strength characteristics in different way. It shows that punching is not sufficiently investigated and additional experimental and theoretical research is required.
Performed analysis of models makes it possible to distinguish the 2 model types: I type -failure takes place due to shearing the compression zone by the principle stress (Broms 1990;Halgren 1998;Theodorakopoulos, Swamy 2002); II type -failure takes place due to tension stress in diagonal section (Georgopoulos 1989, Menétrey 2002.
Presented review makes it clear that the models referred to the 2 nd group are less realistic. Models referred to the 1 st group reflect the real failure mode much better.
Most authors punching phenomenon consider as a plane problem, i.e. the same principles as for the strength of diagonal section for RC beam without shear reinforcement are applied. Then normal and diagonal cracks with the action of the principle stresses progress towards the compression zone and failure occurs when the compression zone is destructed under a certain combination of compression and shear stresses. Actually, the compression zone is under the action of tri-axial stress state but most authors neglect it. It is neglected in almost all design codes and recommendations.
Steel fibres in punching slab zone not only change interrelation between stresses of different types but also their distribution due to the influence of difference in strain properties and their character in elastic plastic structure behaviour stage. Thus, it is important to create such analytical model that would allow complete evaluation influence of steel fibres in determination of punching strength for RC superstructure deck slabs and foundations of bridge.

Diagram of analytical model and assumptions
Analytical model selected on the basis of investigation in strength and deformation properties of SFRC (Šalna, Marčiukaitis 2007) and behaviour of common slabs under punching (Broms 1990;Halgren, 1998;Theodorakopoulos, Swamy 2002) and according to general principles of mechanics is shown in Fig. 2.
For creation, analysis and investigation of the model the following assumption were made: 1) tension and compression stress strain diagrams for the materials are of trapezium shape (Fig. 3); 2) hypothesis of plain sections is valid; 3) in determination of the neutral axis location at failure the ultimate values of concrete strain are used; 4) diagram of normal stress in the compression zone -trapezium, that for tangential stress -in relationship with the normal stress; 5) failure occurs when compression zone of the slab is cut by the principle stress at an angle coinciding to the direction of the said stress. Trapezoidal normal stress diagram is taken for determination of the compression zone. Investigations performed by different authors (Maalej, Li 1994) point out that application of trapezoidal σ-e diagrams in compression and tension zones for SFRC gives results that show better Diagram for determination of neutral axis location for SFRC member is shown in Fig. 3. Stress diagrams for compression and tension zones assumed trapezoidal where elastic tension and compression fibre-concrete strains e sfrc,t,el , e sfrc,c,el correspond to variable stress parts are represented by and , while constant ones -correspondingly by and . Here n sfrc and l sfrcelasticity and plasticity coefficients for fibre-concrete.
l sfrc in tension and in compression are determined by empirical formulas obtained using results of experimental investigations. When steel fibres are with bends: , where l c and l t -plasticity coefficients for concrete in compression and tension. According to investigations , since it is assumed that concrete elasticity module in tension and compression are equal.
Ratios between the average stress value and the max one in the diagrams, below referred to as stress diagrams completeness coefficients: . ( Stresses in compression and tension zones corresponding to e sfrc,c and e sfrc,t strains related to tension reinforcement stress : , where . (4) Then resultants acting in tension and compression of SFRC member are: . (5) When stresses and coefficients of their diagrams for the fibre-concrete are known it is simple to determine their resultants and to calculate compression zone depth from conditions of equilibrium for these resultants.

Analysis of normal and tangential stress diagrams in the compression zone
Analysis of publications revealed that different normal and tangential stress diagrams for principle stress calculations are used by the most of authors (Fig. 2). For example, Choi el al. (2007) apply cubic parabola for normal stress and for tangential one -trapezium, Shertwood et al. (2007) -trapezium for normal stress and square parabola -for tangential one, Zink (1999) -triangle for normal stress and cubic parabola -for tangential one. Nevertheless, trajectories of the principle stress depend very much on selected stress diagrams. Angle of the principle stress trajectory variation in relation to various selected combinations of normal and tangential stress diagrams is presented in the Table 1. When for normal stress trapezoidal diagram in relation to concrete plasticity coefficient (its completeness coefficient w σ = 0.8−1) and for tangential either trapezoidal or triangle diagrams (w σ = 0.5−1) are assumed, the results apparently differ. Between extreme completeness coefficient values, angle of the principle stress trajectory vary from α = 23 up to α = 57.
However, analysis of results of experimental investigations performed by Choi et al. (2007), Shertwood et al. (2007), Sharma (1986), Tuchlinski (2004) and Zink (1999) revealed that completeness coefficient for tangential stress diagram on average w t = 0.65−0.75, which corresponds to the smaller spread of the principle stress trajectories angle: 33 −45. These values are commonly applied in classical theories and they generally agree with the experimental investigations of the said authors. Relation of these results with obtained experimental results and with idealized nor- Fig. 3. Strain and stress diagrams in normal section of SFRC slab with allowance for plastic strain of concrete in tension and in compression mal stress diagram expressed via plasticity coefficient gives completeness coefficients for normal and tangential stress diagrams: .
(6) Tangential stress diagram coefficient value determined in such way by formula varies within the limits of w t = 0.6−0.75. It conforms to the completeness coefficient for diagrams described by square and cubic parabolas.

Principle stress and area of its action in the compression zone
According to proposed punching diagram presented in the Fig. 2 the principle stress depends on normal and shear (i.e. tangential) stresses.
Following the rule that the ultimate compression stress σ u equals to the concrete compression strength f c and the ultimate shear (tangential) stress at compression t u = 0.5f c , applying classical strength of materials formulas principle stress and angle its trajectory is expressed by: . (7) Diagrams for determination of principle stress action in the compression zone area are given in Fig. 4.
When base and slant altitude of lateral surface for truncated pyramid and generatrix of lateral surface for truncated cone are expressed via x and α area of principle stress action in the compression zone may be determined using simple geometrical formulas: for circular columns , for square columns , where a -cross-sectional dimension of square column or cross-sectional diameter of circular column.
Punching force F u is determined assuming that failure takes place due to cutting the compression zone by the principle stress σ 1,2 in the area A: . (10)

Method of investigation
Experimental investigations of punching strength for fibre-concrete slab zone around a column are not many. More extensive data of the said investigations are presented by Haralji et al. 1995;Swamy, Ali 1982;Urban 1984;Раб���в�� 2004. Small amount of tests may be explained by great test materials and labour consumption and by complexity.
Thus experimental investigations giving opportunity to assess punching failure character for SFRC slabs more deeply and to verify the theoretical model were performed.
Four RC slabs were produced and tested: three of them were additionally reinforced by steel fibres and one was control slab. Amount of steel fibres was selected as variable in the experiment while concrete class and longitudinal reinforcement were constant values ( , Table 2). The experimental slabs were concreted in two series: I series -slabs reinforced by 1% and 1.5% of steel fibres; II series -slabs reinforced by 2% of steel fibres and the control concrete slab.
Cement of Cem 42.5R, sand of (0-4 mm), gravel of (4-16 mm), water and plasticizer were used for the concrete mix. Concrete properties presented in the Table 3. Slab zone at distance of 4d from the column face was additionally reinforced with steel fibres. Steel fibres of "Metalproducts" MPZ50/1 (product No. 1010) with its (f y = 1100 MPa) was used. Three different volume fractions of steel fibres for slabs were applied: 1% -78.5 kg/m 3 , 1.5% -117.8 kg/m 3 , 2% -157 kg/m 3 . They were tested in a special test stand. The slabs are supported on 10 mm thick support along the whole their contour. Cement and sand mix was used to make the supports even. Hydraulic jack of 1000 kN was used for slab loading. Diagram of the test presented in Fig. 5.   Slab deflection at its centre was measured during the test with allowance for movements at corners, longitudinal and transverse (radial) strains of tension and compression zones were measured as well. Slab tension zone strains were measured by mechanical (150 mm base) and by electrical resistance (50 mm base) strain gauges. Mechanical gauges for longitudinal and radial strains were arranged in 3 specific zones: at d, 2d, 3d distances from column face in perpendicular and diagonal directions. Compression slab face stains in d zone were measured only by electrical resistance strain gauges. Arrangement of mechanical and electrical resistance strain gauges during the test is presented in Fig. 6.

Results of experimental investigations
Analysis of available experimental investigation results revealed that steel fibres affect not only the carrying capa-   (Harajli et al. 1995;Kutzing, Konig 2000;Urban 1984). It was determined by comparison deformation properties of control and SFRC slabs that they differ substantially. For example, slab deflection at its centre at the ultimate load for SFRC slab is 2.8 times (Fig. 7) and at failure -more than 3 times greater than that for the control slab. Moreover, failure modes are very different -the control slab failed suddenly, almost unexpectedly (Fig. 8a) while SFRC slabs failed plastically. When the ultimate load was reached, further loading resulted in decrease of the load carried by the slab and intensive growth of deflections (Fig. 8b). In fact there was no sudden failure.
Analysis of slab deformation during loading and tension zone failure mode revealed evident difference. The first radial crack opened almost at the same load in both slabs: at 0.14F u -in the RC slab without steel fibres and at 0.14F u -0.17F u -in SFRC slabs. Development of cracks is similar in both radial and tangential directions. For example, all cracks in radial direction at distance d from the column face appeared at 0.1-0.4F u , at 2d distanceat 0.3-0.6F u , while in tangential direction -at 0.25-0.4F u and 0.3-0.6F u respectively. However, cracks in SFRC slabs are distributed much closer to each other. Obvious development of tangential crack width in the zone of 0.5d from the column face in the concrete slab is observed when the load is razed from 0.6-1.0F u . In SFRC slab it develops at greater distance -0.7-1.2d. Distinct formation of the punching cone starts, critical future punching perimeter becomes clearly visible.
These factors altogether show that formation of compression zone depth as well as direction of the main shear stress and its value depends on steel fibres. That is why these factors are assessed in the proposed model.
Saw-cuts of the slabs were made after their tests (Fig. 9). It is seen from the transverse saw-cuts of the slabs that the punching cone in concrete slabs formed at the angle of about 27� (2d from the column face) while in SFRC slabs at smaller angle -21�-23�(2.4-2.6)d. Diagonal crack develops up to the compression zone which is cut by the principle stress at different angle as well. Moreover, different depth of compression zones is clearly seen. Compression zone depth in SFRC slab is substantially greater than that in common RC slab.

Analytical model compared with experimental results and models of other authors
Accuracy of proposed model was investigated first. Therefore, comparison of experimental values presented by various authors ( Fig. 10) with theoretical values obtained according to the proposed model was made.
Comparison of experimental values presented by various authors with theoretical values according to the proposed model in graphical form is given in Fig. 10 for accuracy assessment of the model for the case with steel fibres. Steel fibres of various geometrical, strength and different anchorage (with full and half bends, wavy and smooth shape) properties were used for comparison as well. It was determined from these data that the average ratio between experimental and theoretical values according to the proposed model standard deviation σ X -0.08 and coefficient of variation v -7.0%. According to these statistical data for the proposed model with steel fibres systematic error equals to μ R = 1.12 and the random one σ R = 0.08. Analytical model was compared with different experimental data of other authors (Fig. 10). Specimens of their tests differ by geometrical parameters, concrete strength and longitudinal reinforcement quantity: f c = 14−120 MPa, f y = 330−706 MPa, r l = 0.33−3%, d = 99−476 mm.
Parallel to x-axis regression straights joining related points (Fig. 11) shows that the proposed model is not sensitive to various factors, such as slab geometrical characteristics, steel fibres type, anchorage properties etc. Раб���в�� (2004) method is very close to the proposed one since it is based on the same design principles of composites, but plastic strains and influence of longitudinal reinforcement on punching strength are not taken into account and larger deviations are obtained. Distinctly stands out Harajli et al. (1995) method that demonstrates clearly that not for all cases punching strength analysis using the same empirical relationship for evaluation of steel fibres influence on carrying capacity can be carried out. Results presented in Table 4 shows that proposed model is substantially more accurate than other known models.

Conclusions
Model for punching strength analysis of bridge flat slab superstructure deck and foundation slab is proposed which allows evaluation of all the most important factors influencing punching strength. It is proven that these factors are concrete strength, amount of longitudinal and steel fibres, and its strength and anchorage properties. The model gives opportunity to take into account SFRC strength not only in compression zone but also in tension zone as well that is an impossible using model with longitudinal reinforcement only.
Proposed method for evaluation plasticity properties of SFRC and for selection diagrams of normal and tangential stresses acting in slab tension and compression zones during punching and area on which the principle stresses act taking in to account plastic strains of SFRC.
Model accuracy verified comparing with experimental results. Accuracy figures of the proposed model obtained comparing experimental results with analytical ones are: average of the mean values m x = 1.12, their square deviations σ x = 0.08 and variation coefficient n = 0.07.
Behaviour under the load of SFRC slabs is substantially different from that of slabs reinforced with steel bars. Their cracking starts at larger loads, spacing of cracks is less, slab failure is of plastic type. At failure slab deflects 3 times more than that without steel fibres.