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1. INTRODUCTION

The innovative approach to shear design of the reinforced and prestressed concrete members is proposed by First Complete Draft of fib Model Code 2010 [3]. The substantial adjustment in these shear provisions is the inclusion of the concept of a “Level of Approximation”, according to which in the determining structural concrete member shear resistance a different levels of approximation may be regarded. The levels (I, II, III and higher) mutually differ in the complexity of the applied methods, the effort, the level of detailing, the accuracy of the obtained results and the precision of evaluation. The shear design set of the equations for Level III Approximation is derived from Modified Compression Field Theory (MCFT) [7], after introduction series of additional assumptions and simplifications, which are in detail elaborated in references [6] and [2]. The another simplified shear design approach (based on similar assumptions, but with different final results) based on tabulated values of two main parameters is previously developed and incorporated in AASHTO LRFD Bridge Design Specifications (since 2nd Edition 1998.). This approach was continuously changed and improved, and still is current (5th Edition 2007. reference [1]).

Generally, the shear design according to both approaches is based on the proposed procedure for determining average longitudinal strain in the beam cross section under certain combination of the applied design sectional forces. Besides that, the condition that the flexural capacity of the longitudinal reinforcement is sufficiently high to allow a shear failure must be fulfilled. The procedure for determining average longitudinal strain in the beam cross section under certain combination of the applied design sectional forces is presented in the references [3], [2], and [1]. The undimensional relationship between shear resistance coefficients, developed in reference [5], incorporates code proposed limitations related to minimal shear reinforcement ratio and maximal shear resistance value. Another option for practical shear design of beam cross sections recommended in reference [1] is construction of the shear-moment interaction diagrams. The procedure for construction of the shear force-bending moment interaction diagrams according to both previously mentioned approaches and intended for use in shear design of the reinforced concrete beam cross sections exposed to compressive or tensile axial force, is presented in reference [6].

The transversal reinforcement area, distribution and quality, and design axial force are variable parameters on the interaction diagrams, while beam cross section geometric characteristic, concrete class, longitudinal steel area, distribution and quality must be previously chosen and fixed. The vertical stirrups area and spacing, and corresponding steel quality are described with one parameter which has physical meaning as design stress carried by provided vertical stirrups. The influence of different level of beam design axial force on the interaction diagram shape is studied and obtained results of the analysis are described in this paper.

2. THE SHEAR-MOMENT INTERACTION DIAGRAM

The basic model of shear resistance of reinforced concrete beams is presented on Fig. 1, which shows a free body diagram that follows the diagonal shear crack and cuts the flexural reinforcement, stirrup legs, and flexural compression region.

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Figure 1. Basic model of the reinforced concrete beam shear resisting mechanism

The bending moment at the critical shear location is carried by a force couple between the flexural compression force in the concrete (C) and the tensile forces in the longitudinal reinforcement (T). The acting shear force (VEd) is assumed to be carried by vertical forces in the yielding stirrup legs that cross the diagonal crack (VRd,s), shear stresses on the crack itself (vc) commonly called aggregate interlock stresses and shear stresses in the flexural compression region of the beam.

The shear area is the web width (bw) multiplied by the flexural lever arm (dv). This assumption has been selected because it is closer to the reality of how shear is resisted by a section. The flexural lever arm (dv), also called effective shear depth, can be taken as the distance, measured perpendicular to the neutral axis, between the resultants of the compressive and tensile forces due to flexure, but need not be taken less then (0.9d) where (d) is section effective depth i.e. distance from the extreme compression fiber to the centroid of longitudinal tension reinforcement in section. Additional assumption in the shear model is that the aggregate interlock resistance of the complex crack geometry may be estimated at only one depth in the beam and that this can represent the entire crack surface. The shear force acting on the section shear area is (VEd) and the shear stresses (v) are assumed to be uniformly distributed over previously defined shear area.

Figure 2. The distribution of section internal forces and strains in the beam cross section

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The design shear stress (v) is then defined as:

VW

Ed

db

Vv = (1)

The second important parameter is average longitudinal strain (εx) at a given depth of the beam section. The depth in the cross section at which this strain is calculated corresponds to center of assumed shear area and approximately represents mid-depth of the cross section. As the strain in flexural compression tends to be small due to the high stiffness of concrete in compression, the longitudinal strain (εx) is conservatively approximated as one half of the strain in the flexural tensile reinforcement.

22ttc

x

ε≈

ε+ε=ε

(2)

The Fig. 2 shows a small part of a beam resisting external sectional forces on the left side with the assumed internal force mechanism resolved on the right side including diagonal compressive stresses (f2) between the cracks.

The force (T) in the tension reinforcement is converted to a strain using the relationship shown on the right side in Fig. 3 and the longitudinal strain in bottom chord (εt) is:

ss

EdEdvEdt EA

N5.0cotV5.0d/M +θ+=ε (3)

Figure 3. The distribution of section internal forces and relationship (T-εt) in bottom tension chord for reinforced concrete beam cross section

The equation for determining the longitudinal strain parameter (εx), after adoption of the approximation (0.5cotθ≈1) and approximation (2) is obtained as:

ss

EdEdvEdx EA2

N5.0Vd/M ++=ε (4)

where (MEd) is the applied design moment, (NEd) is the applied design axial force, (As) is the area of flexural reinforcement and (Es) is reinforcement modulus of the elasticity.

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The design moment and shear (MEd and VEd) should always be taken as positive quantities and (NEd) is taken as negative for compression and positive for tension. In using equation (4) (MEd) shall not be taken less than (VEd·dv), or the following inequality must be fulfilled.

v

EdEd d

MV ≤ (5)

The previous limitation is postulated in the references [3], [2] and [1]. At minimal moment locations, such as the inside edge of the bearing area of simple beam end supports, the tensile force that longitudinal reinforcement is expected to resist is calculated on bases of sectional forces for the section at distance (dv) from the face of support. In calculating the tensile resistance of the longitudinal reinforcement a linear variation of the resistance over the development length is assumed, i.e. the area of bars which are terminated less than their development length from the cross section under consideration shall be reduced in proportion to their lack of full development.

The parameter (εx) indicates how the demand on the longitudinal reinforcement compares with the quantity of reinforcement provided. The assumption (0.5cotθ≈1) is a conservative because it will tend to slightly overestimate the impact of shear on the force (T) for the permissible range of (θ). With this simplification, the longitudinal strain equation (4) no longer depends on the angle of principal compression stresses (θ) and thus no longer requires iteration to evaluate.

The shear resistance of a beam is determined according to:

Eds,Rdc,RdRd VVVV ≥+= (6)

where (VRd) is design shear resistance, (VRd,c) is design shear resistance attributed to the concrete, (VRd,s) is design shear resistance provided by shear reinforcement and (VEd) is design value of shear force.

The shear resistance is limited to value:

θ+α+θ

γ=≤

2vw

c

ckCmax,RdRd cot1

cotcotdb

fkVV (7)

where (fck) is characteristic value of cylinder compressive strength of concrete, (γc) is partial safety factor for concrete material properties, (θ) is angle between web compression stresses and the beam axis and (α) is inclination of stirrups to the beam axis. For a Level III Approximation a value (θ=45°) shall be inserted in previous equation.

Coefficient (kc) for Level III approximation is determined as:

55.0f

3055.0k

3

1

ck

C ≤

= (fck in MPa) (8)

The design shear resistance attributed to the concrete (VRd,s), for members with amount of the shear reinforcement greater then minimal, can be determined as:

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The obtained VEd-MEd-NEd surface defines boundary of the region with acceptable ultimate limit state design values of the cross sectional forces. The developed procedure of shear force–bending moment–axial force interaction diagram construction is easily programmable and may be performed in spreadsheets. Then with the one interaction diagram great number of load cases, corresponding to part of beam can be considered.

4. CONCLUSIONS

The procedure for construction of the shear force – bending moment –axial force interaction diagrams, intended for use in shear design of the reinforced concrete beam cross sections, based on set of the provisions and equations recommended by fib Model Code 2010, for approximation level III, is proposed. The set of the equations analyzed in the paper is related to reinforced concrete beams with well detailed and distributed shear reinforcement (stirrups) primarily made of deformed bars without upper limitation on the applied concrete class. The vertical stirrups amount (area and spacing) and steel quality are described with one variable parameter which have physical meaning as design stress carried by provided vertical stirrups. It is variable interaction diagram parameter, while beam cross section geometric characteristic, concrete class, longitudinal steel area, distribution and quality must be previously chosen.

As can be seen axial compression considerably increases ultimate shear capacity of the section in the absence of high flexural loads. Axial tensile forces result in a significant decrease in shear strength, as is expected.

5. REFERENCES

1. AASHTO LRFD Bridge Design Specifications, 4ed Edition, American Association of State Highway and Transportation Officials, 2007. p. 1512.

2. Bentz, E.C., Collins, M.P.: Development of the 2004 Canadian Standards Association (CSA) A23.3 Shear Provisions for Reinforced Concrete, Canadian Journal of Civil Engineering, NRC Research Press, Vol.33 (5), June 2006., pp. 521-534.

3. International Federation for Structural Concrete (fib): „fib Model Code 2010“, First Complete Draft–Volume 2, Fib Bulletin 56, April 2010., p. 311.

4. Popović, D.B.: Actual Approach to Shear Design of the Prestressed and Reinforced Concrete Beams“, Proceedings of International Symposium about Research and Application of Modern Achievements in Civil Engineering in the Field of Materials and Structures, Society for Materials and Structures of Serbia, XXV Congress-Tara, October 2011., pp. 219-226.

5. Popović, D.B.: The Shear-Moment Interaction Diagrams of the Reinforced Concrete Beams, Proceedings of 4th International Conference "Civil Engineering – Science and Practice", GNP 2012, Faculty of Civil Engineering, University of Montenegro, March 3 2012., pp. 297-305.

6. Rahal, K.N., Collins, M.P.: Background to the General Method of Shear Design in the 1994 CSA-A23.3 Standard. Canadian Journal of Civil Engineering, NRC Research Press, Vol. 26 (6), 1999., pp. 827–839.

7. Vecchio, F.J., and Collins, M.P.: The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear, American Concrete Institute Journal, Vol.. 83 (2), Mart-April 1986., pp. 219–231.