Shear assessment of compression flanges of structural concrete T-beams

Abstract

In T-beams the force transfer from the web into the flange has to be studied. The general design procedure is based on a strut-and-tie (or a stress field) model which comprises spreading compressive and transverse tensile forces. As is known, strut-and-tie models represent the force flow within a structural member at ultimate. This procedure is sufficient for design purposes and in general, leads to safe results. For the assessment of a structure it may be worthwhile to improve the accuracy. For this purpose both web and flange have to be looked at more in detail. An advanced method for the analysis of webs in shear is the Generalized Stress Field Approach [1]. This approach can be utilized for treating flanges, where the classical assumptions have to be adapted; in particular by considering the strain dependence of the concrete compressive strength and thus, defining a representative strain value. In the present contribution background and details of these aspects are given, and the corresponding calculation procedure is described. Theoretical results are compared with experimental data and show a reasonably good agreement. However, as the number of sufficiently documented tests is very limited no concluding findings are attained.

Keywords

concrete structures / structural assessment / stress field analysis / shear

Introduction

The spread of compressive forces from the web of a beam into its flanges is usually studied with help of a strut-and-tie model or a stress field analysis [ 2– 4]. For the design of a structural member this leads to a sufficiently safe result. However, for the assessment of a structure it may be worthwhile to improve the accuracy and to develop a more detailed method. Such a method is presented in the following. It is based on a mechanical-physical model, the Generalized Stress Field Approach [ 1]. This approach employs a stress field analysis and combines the limit analysis method with compatibility considerations. It is worth noting that the Level II Approximation for shear design of webs according to fib Model Code 2010 [ 4] is based on this approach. In that case, a cross-sectional design procedure for the so-called control section at a certain distance from the support or point load is examined. For the adaptation of this approach, the distribution of longitudinal stresses and strains in the flange has to be determined and failure criterion established. In the following, background and details of these aspects are given, and the corresponding calculation procedure is described.

Generalized stress field approach

A discontinuous stress field represents a model of the stress state at ultimate. For the analysis of a girder, nodes and fans are presupposed at locations where concentrated loads are introduced, and parallel stress-bands (or fans) in the remaining parts of the structure. Within the framework of the Generalized Stress Field Approach [ 1] it is presupposed that the strain at the mid-depth of the beam is representative for the state of strain in the web; this assumption has been adopted by the fib Model Code 2010 [ 4] shear provisions. Simplifying, quite often the strain at the axis of the compression chord (or flexural compression zone) is taken as zero (Fig. 1).

Figure 1 shows the free body diagram of a centered fan which represents the equilibrium system of the applied and internal forces and the assumed distribution of the longitudinal strains. In the figure, z denotes the effective shear depth and the lever arm of internal forces, respectively, F_c the force in the compression chord, F_s the force in tension chord, ϵ_s,x the strain in the longitudinal reinforcement and ϵ_m,x the longitudinal strain at the mid-depth of the effective shear depth. Particularly, the uniformly distributed load q is equilibrated by the stirrup forces a_sw f_y (with a_sw denoting the cross-sectional area of a stirrup per unit length, i.e., A_sw/s_w). Simplifying, it is assumed that within the fan the directions of principal stresses and strains coincide; hence, the gray solid lines in Fig. 1 depict the stress and strain trajectories, respectively.

The generalization of the plasticity based stress field analysis follows through the implementation of specific limits for the inclination of the trajectories θ and the definition of the effective concrete strength as a function of the strain state [ 1]. The lower limit θ_min corresponds to stirrup rupture and/or crack sliding failures; the upper limit θ_max is defined by the transition to the strain state in which the stirrups do not reach yielding, and the failure occurs by pure crushing of the web concrete. In [ 1] it is proposed to determine these limits on the basis of the kinematics of the web element and the Tension Chord Model [ 5]. These considerations yield the expression

(1) {\mathit{\theta }}_{min,max}=\arctan {\left({\displaystyle \frac{{\mathit{ϵ}}_{m,x}-{\mathit{ϵ}}_{\mathrm{2},m}}{\mathit{\kappa }·{\mathit{ϵ}}_{s,max}-{\mathit{ϵ}}_{\mathrm{2},m}}}\right)}^{\mathrm{1}/\mathrm{2}}.

In Eq. (1) ϵ_2,m denotes the minimum principal strain at the mid-depth of the web and can be taken as -0.002 (or the peak strain of concrete in compression, respectively). Minimum and maximum inclinations could more accurately be determined with help of the Cracked Membrane Model [ 6]; the application of Eq. (1) is much easier and the results are sufficiently accurate and thus, used for the case treated in this paper. At θ_min the stirrup strains are limited to κ·ϵ_su where ϵ_su denotes the strain at rupture of the steel in the cracks. At θ_max the strain κ·ϵ_s,max is given by κ·ϵ_sy, ϵ_sy denoting the yielding strain of the steel. As the relevant bond coefficients κ depend on the ductility characteristics of the steel κ can be computed or approximately be taken as κ = 0.25 for the lower limit and κ = 0.8 for the upper limit of θ. For practical applications instead of Eq. (1) even simpler relations can be used, see e.g., Ref. [ 4].

As mentioned above, the compressive strength of the web concrete depends on the state of strain which can be determined with the help of Mohr's circle of strains. For instance, the principal strain ϵ₁ is found to

(2) {\mathit{ϵ}}_{\mathrm{1}}={\mathit{ϵ}}_{x,m}+\left({\mathit{ϵ}}_{x,m}-{\mathit{ϵ}}_{\mathrm{2},m}\right)\cdot {\cot }^{\mathrm{2}}\mathit{\theta }\mathit{.}

Based on experimental findings and theoretical considerations [ 1] the effective concrete compressive strength f_ce can be defined as

(3) f_{ce}={\mathit{\eta }}_c\cdot f_c\cdot {\left({\displaystyle \frac{\mathrm{30}}{f_c}}\right)}^{\mathrm{1}/\mathrm{3}}\le {\mathit{\eta }}_c\cdot f_c\mathrm{with}{\mathit{\eta }}_c={\displaystyle \frac{\mathrm{1}}{\mathrm{1.2}+\mathrm{55}\cdot {\mathit{ϵ}}_{\mathrm{1}}}}.

In Eq. (3) f_c denotes the cylinder strength and η_c the reduction factor accounting for strain influences; additionally, the compressive strength is reduced for f_c>30 MPa due to the more brittle behavior of such concrete types.

Expressed for the shear stresses τ, the resistance provided by the stirrups τ_R,s and the limit due to crushing of the web concrete τ_R,max are obtained with Eq. (4) and Eq. (5):

(4) {\mathit{\tau }}_{R,s}={\mathit{\rho }}_z\cdot f_y\cdot \cot \mathit{\theta },

(5) {\mathit{\tau }}_{R,max}=f_{ce}\cdot \sin \mathit{\theta }\cdot \cos \mathit{\theta }.

Here, ρ_z is the geometrical reinforcement ratio and f_y the yield strength of reinforcing steel. The inclination of the trajectories for ultimate state θ can be obtained from equalisation of Eq. (4) and Eq. (5). This leads to following expression for the so-called web crushing criterion with ω_z = ρ_z f_y/f_c as the mechanical stirrup reinforcement ratio.

(6) \mathit{\theta }=\arcsin \left(\sqrt{{\displaystyle \frac{{\mathit{\omega }}_z\cdot f_c}{f_{ce}}}}\right).

Application of generalized stress field approach to flanges

Flanges of beams are membrane elements and can hence be treated as (in most cases orthogonally reinforced) concrete panels. In the following and in the first approximation, it is assumed that the strength definition for webs according to [ 1] can also be used for flanges. To substantiate this assumption, comparisons with other strength definitions; based on panel tests were carried out, where the test results have been adopted from Ref. [ 7]: included are tests from Kaufmann [ 7], Vecchio and Collins [ 8, 9], André [ 10], Khalifa [ 11], Kirschner and Collins [ 12], Marti and Meyboom [ 13], Ohmori et al. [ 14], Pang and Hsu [ 15], Collins and Porasz [ 16], Zhang and Hsu [ 17], and Belbarbi and Hsu [ 18]. In these test series the concrete compressive strength varies from 11.6 to 103.1 MPa. For comparison, the strength definitions of the well known proposals of Belbarbi and Hsu [ 18] (Eq. (7)), Kaufmann [ 7] (Eq. (8)) and Vechio and Collins [ 19] (Eq. (9)) are used here.

(7) f_{ce,Hsu}=f_c\cdot {\displaystyle \frac{\mathrm{0.9}}{\sqrt{\mathrm{1}+\mathrm{250}\cdot {\mathit{ϵ}}_{\mathrm{1}}}}},

(8) f_{ce,Vecchio}=f_c\cdot \underset{\le \mathrm{1}}{\underbrace{\frac{\mathrm{1}}{\mathrm{0.8}+\mathrm{170}\cdot {\mathit{ϵ}}_{\mathrm{1}}}}},

(9) f_{ce,Kaufmann}=\underset{\le f_c}{\underbrace{\frac{f_c^{\mathrm{2}/\mathrm{3}}}{\mathrm{0}·\mathrm{4}+\mathrm{30}\cdot {\mathit{ϵ}}_{\mathrm{1}}}}}.

The strength definitions according to Eq. (3) and Eq. (9) not only depend on the principal strain ϵ₁ but also on the concrete strength f_c. Therefore, in Fig. 2 the upper and lower limits for the available tests are specified. To visualize of the influence of the concrete strength the tests were subdivided into three categories (s. Figure 2: f_c≤30MPa, 300 MPa<fc≤60 MPa and fc ≥ 60 MPa).

The evaluation of the tests shows a large scatter. However, it can be concluded that a formulation independent of the concrete strength is not sufficient. In particular, for most of the data and concrete strength categories Eq. (3) shows a good agreement and seems to be justified; only for a small number of results and the highest concrete strength category Eq. (9) is more suitable. Due to this assessment and to obtain a consistent calculation procedure for both, the web and the flange, Eq. (3) is used in the following. It is worth noting that failure conditions and types might differ from those in the web; as there is only a very small number of sufficiently documented tests this stipulation is taken as provisory.

Force introduction into flanges

Because of the unknown location of the failure, cross-sectional design as carried out for webs is not directly applicable to flanges. Here, a more refined approach that considers the load distribution in the flanges and the variable strength of the concrete struts is necessary. The transfer of forces from the web into the flange depends on many factors, e.g. the static system, the loading stage, the ratios of longitudinal reinforcement and stirrups in the web and the ductility characteristic of reinforcing steel. To be able to account for all these influences, a calculation procedure for an advanced web analysis is developed.

First of all, a stress field is developed i.e., the web is subdivided into compressive bands; background information about this step may be found e.g., in Refs. [ 20, 21] or [ 22]. In discontinuity (D-) regions the inclinations of the concrete struts θ are given by the geometry of the fan; in Bernoulli (B-) regions the inclination can be determined from kinematics (Eqs. (1) and (6)). The mid-depth strain in longitudinal direction follows from an equilibrium iteration, considering forces due to flexure and assuming a linear strain distribution over the depth. A bilinear stress-strain characteristic for steel and a parabola-rectangle diagram for concrete are used. The average longitudinal strain ϵ_x,m is found at the mid-depth as shown in Fig. 3; c denotes the height of the compression zone and σ_c the concrete stress in longitudinal direction.

Figure 4 shows a centered fan in a D-region of a beam. It is assumed that the shear stress at the top end of the fan τ_w(x) is representative for the force transfer into the flange. In this case, the shear stress at the level of the compression chord force F_c follows from Eq. (10). It is assumed that the position of the reference axis for the bending moment M(x) and the shear force V(x) remains at the mid-depth of the web; simplifying, because of its rather small influence the inclination of the top chord force is neglected. In the case of elastic behavior of steel and concrete, the lever arm of internal forces z(x) decreases with growing moments M(x), and it increases for plastic strains in concrete and steel; Fig. 4 shows the first case.

(10) {\mathit{\tau }}_w\left(x\right)={\displaystyle \frac{V\left(x\right)\cdot \cot {\mathit{\theta }}_w\left(x\right)}{b_w\cdot z\left(x\right)\cdot \cot {\mathit{\theta }}_{fa}}}.

In Eq. (10), θ_w(x) denotes the inclination of the trajectories which vary over the length of the beam, b_w the width of the web and θ_fa the lowest inclination of the fan at the discontinuity line. In B-regions the inclination θ_w(x) of the trajectories is found from Eq. (6) and Eq. (1). This implies that these equations are valid for all load stages. In this case, the shear stress at the top end of the fan τ_w(x) can be found with Eq. (11). The supposed stress field for B-regions is shown in Fig. 5. In Fig. 5 Δ(x) is the shift and the gray dotted line marks the reference axis.

(11) {\mathit{\tau }}_w\left(x\right)={\displaystyle \frac{V\left(x\right)\cdot \cot {\mathit{\theta }}_w\left(x\right)}{b_w\cdot z\left(x\right)\cdot \cot {\mathit{\theta }}_w\left(x+\mathrm{\Delta }\left(x\right)\right)}}.

With the help of Eq. (10) and Eq. (11) the shear stress τ_w(x) can be determined for every location at the mid-depth along the beam i.e., for every given coordinate x by computing the inclination θ_w(x) and the lever arm of internal forces z(x). For the introduction of the web forces into the flange the shear stress at x has to be shifted to the location x + Δ(x). Additionally, different thickness of web and flange and the spreading portion of the compression chord force have to be observed. For the former, the shear stress in the web has to be multiplied by the ratio of the two values. For the latter and usually more important, the compression chord has to be studied more in detail; therefore, the effective flange width, e.g., according to EC2 [ 2] (Fig. 6) is used.

The distribution of compressive stresses and the resulting forces are known from equilibrium iteration. The portion of the compression chord force that is spread into each half of the flange can be expressed by the reduction factor λ(x) according to Eq. (12):

(12) \mathit{\lambda }\left(x\right)={\displaystyle \frac{F_{c,fl}\left(x\right)}{F_c\left(x\right)}}.

Finally, the shear stress in the flange τ_E,fl(x) can be found as follows:

(13) {\mathit{\tau }}_{E,fl}\left(x\right)={\mathit{\tau }}_w\left(x\right)\cdot {\displaystyle \frac{b_w}{h_f}}\cdot \mathit{\lambda }\left(x\right).

Furthermore, it is assumed that the longitudinal strain at mid-depth of the flange ϵ_m,x,fl is representative for the analysis (Fig. 7). The reference axis is defined at the transition line of the web and the flange (dotted line in Fig. 8).

With the longitudinal strain ϵ_m,x,fl and the strength definition according to [ 1] the resistance of the flange τ_R,s,fl and τ_R,max,fl can be, analogously to the web design, expressed as

(14) {\mathit{\tau }}_{R,s,fl}={\mathit{\rho }}_y\cdot f_y\cdot \cot {\mathit{\theta }}_{fl},

(15) {\mathit{\tau }}_{R,max,fl}=f_{ce,fl}\cdot \sin {\mathit{\theta }}_{fl}\cdot \cos {\mathit{\theta }}_{fl}.

Eventually, this procedure represents the Generalized Stress Field Approach that incorporates a description of the flow of the internal forces i.e., a stress field; the latter has been developed for design purposes by Muttoni et al. [ 22]. Figure 8 shows this stress fields for the Beam Q2 from the test series of Bachmann [ 3]. The variation of the lever arm of internal forces is relatively small, which is because the longitudinal reinforcement remains in the elastic range.

The spread of the forces into the flange is equilibrated by the transverse stirrup reinforcement and the longitudinal compression. In Fig. 8 the minimum mechanical reinforcement ratio ω_min the required mechanical stirrup reinforcement ratio ω_y,req which varies along the beam length is shown. This reinforcement ratio can be found with the help of Eqs. (13) and (14). It is worth noting that there is considerable shift to the location at which reinforcement is required; this location is defined by equilibrium. From the comparison of the available and the required reinforcement, the ultimate load can be obtained in an iterative procedure.

Comparison with Eurocode 2 provisions

To examine the potential of the presented approach, a comparison of the proposed analysis procedure with the design method according to Eurocode 2 (EC2 [ 2]) is carried out. According to EC2 shear stresses τ_E,EC2 are obtained from Eq. (16), where ΔF_c denotes the change of the chord force over the length Δx which is half the distance from maximum bending moment to the point of zero moment. Usually, the lever arm of internal forces z may be estimated as 0.9·d, with d being the effective flexural depth. To obtain the change of the top chord force in one half of the flange, the total force is split up in proportion to the related areas. These simplifications lead to Eq. (17), A_c,fl denoting the area of half the flange and A_c the total area of the compression cord.

(16) {\mathit{\tau }}_{E,EC\mathrm{2}}={\displaystyle \frac{\mathrm{\Delta }{F}_c}{h_f\cdot \mathrm{\Delta }x}},

(17) {\mathit{\tau }}_{E,EC\mathrm{2}}={\displaystyle \frac{{\displaystyle \frac{M_{max}}{\mathrm{0.9}\cdot d}}\cdot {\displaystyle \frac{A_{c,fl}}{A_c}}}{h_f\cdot \mathrm{\Delta }x}}.

The resistance of the flange can be obtained analog to Eqs. (14) and (15) even if the definitions of strut inclination and concrete strength reduction differ from those presented here. Within the limits given by EC2 (Eq. (18))

(18) \mathrm{45}°\le \cot {\mathit{\theta }}_{fl,EC\mathrm{2}}\le \mathrm{26}.\mathrm{5}°,

the inclination θ can freely be chosen, but to enable the comparison it is determined similarly to Eq. (6) by using the strength definition of Eq. (19):

(19) f_{ce,EC\mathrm{2}}=f_c\cdot v=f_c\cdot \mathrm{0.75}\cdot \underset{\le \mathrm{1}}{\underbrace{\left(\mathrm{1.1}-\frac{f_c}{\mathrm{500}}\right)}}.

This strength reduction factor is in accordance to EC2-NA(D) [ 23]. For the comparison the Beam Q1 from the test series of Bachmann [ 3] is studied in detail. Structural geometry, transverse flange reinforcement (stirrups) and ultimate load F_u are given in Fig. 9. The longitudinal reinforcement consists of bars with a total cross-sectional area of 4241 mm²; the web stirrups are diameter 12 mm@100 mm and the concrete cylinder strength is 26.1 MPa. Failure occurred at the interface between web and flange by a combination of concrete crushing and flange shearing off; in the following, this failure mode is denoted as flange crushing failure (FCF). The failure location was near the point load. Because of missing information about material properties, the ultimate strains of the reinforcing steel ϵ_su is taken as 0.05.

For the comparison of both analysis procedures the provided mechanical reinforcement ratio ω_y,prov and the required mechanical reinforcement ratio ω_y,req are shown in Fig. 10 as a function of the longitudinal axis x.

As can be seen in Fig. 10, the proposed model is in good agreement with the test result. Near the point load (x = 2 m) the provided (black dotted line) and the required (black solid line) reinforcement ratio are almost the same at ultimate. Because of the small mechanical reinforcement ratio the lower limit of the inclination θ has to be used for the analysis according to EC2. The kinematic relation of Eq. (1) leads to an inclination in the range of 16° to 27°, depending on the location along the beam. This means that in comparison to EC2 the reinforcement ratio could be reduced. Overall, the more realistic strength definitions of Eq. (3) result in more realistic predictions.

Incidentally, assuming a constant shear stress over the length according to EC2 leads to an excessive demand for reinforcement (gray dotted line in Fig. 10). In this example, the procedure according to EC2 yields a correct result only at the location of maximum requirement; but this is not general. If there had been equal distances between the transverse stirrups, the EC2 results would be even more conservative.

Comparison with test results

To allow for a more general impression of the potential of the analysis procedure presented here, additional tests results were examined. Tests of flanges in pure shear (no transverse bending) are very rare in literature. For validation only the test series of Bachmann [ 3] and Tizatto [ 24] could be used; these tests correspond to the desired load configuration and are sufficiently documented. As for the tests of Bachmann, also for the reinforcing steel used by Tizatto the ultimate strain ϵ_su is assumed as 0.05.

Alternative to what is shown in Fig. 10, the failure load F_calc can be determined; it is found for the location where ω_y,req coincides with ω_y,prov. With that, the ratio F_exp/F_calc is defined, with values>1 indicating a safety margin. The results of this investigation are summarized in Table 1. For comparison, the predictions according to EC2 are given as well.

Even if the number of tests is small, the agreement with the predictions is reasonably good. Particularly, in comparison with the EC2 results, where e.g., for Q1 the ratio F_exp/F_calc of 1.57 lies clearly above the value of 1.11 according to the new model. The occurrence of a flexural failure for Beam Q2 [ 3] is confirmed by the calculation. It has to be noted that for such a small database a statistical evaluation makes no sense and no final conclusions are to be drawn.

Of particular interest are the Beams MT2 and MT3 [ 24]; the predictions are far on the safe side. A more detailed parametric study reveals that the reason for that might be related to the very small amount of transverse reinforcement in flanges: For such cases a significant concrete contribution can be supposed, and could therefore be considered. Up to now, there is no general model that accounts for this portion of the resistance. Also in the Model Code 2010, the concrete contribution is defined rather empirically to form a transition [ 25] between the Generalized Stress Field Approach and the Simplified Modified Compression Field Theory [ 19]. Further research in that field is needed, and also the authors plan to contribute.

Conclusions and recommendations

In this contribution, an advanced method for treating the spreading of compression forces into flanges on the basis of the Generalized Stress Field Approach is presented. The proposed analysis procedure yields more accurate strength predictions compared to the usually used design methods. In the case of structural assessment, the enhanced effort that is needed can be worthwhile. Particularly, the stress and strength distributions along the beam can be studied; this deepened insight in the structural behavior often is especially useful.

For the calculation it is recommended to use the effective flange width according to current codes of practice. This assumption enables the application of the model also for beams with wide compression flanges. In the case of additional transverse bending in the flange, the amount of the respective reinforcement has to be superimposed. For the range of small, medium and high transverse flange reinforcement ratios, the proposed model is expected to yield reasonably safe predictions. Only for very small reinforcement ratios the results might be overly conservative. However, as the number of sufficiently documented tests is very limited no concluding findings are attained up to now.

References

[1] Sigrist V. Generalized stress field approach for analysis of beams in shear. ACI Structural Journal, 2011, 108(4): 479–487

[2] DIN EN 1992–1-1:2011–01: Eurocode 2: Bemessung und Konstruktion von Stahlbeton- und Spannbetontragwerken- Teil 1–1: Allgemeine Bemessungsregeln und Regeln für den Hochbau, CEN, European Standard, 2011

[3] Bachmann H. Längsschub und Querbiegung in Druckplatten von Betonträgern, Beton- und Stahlbetonbau. 1978, 73(3), March

[4] fib Model Code 2010 – Final draft, Volume 2, fib Bulletin No. 66, 2012

[5] Marti P, Alvarez M, Kaufmann W, Sigrist V. Tension chord model for structural concrete. Structural Engineering International, 1998, 8(4): 287–298

[6] Kaufmann W, Marti P. Structural concrete: Cracked membrane model. Journal of Structural Engineering, 1998, 124(12): 1467–1475

[7] Kaufmann W. Strength and deformations of structural concrete subjected to in-plane shear and normal forces. Dissertation for the Doctoral Degree. Report No. 234, Institute of Structural Engineering, Zurich: ETH Zurich, 1998, 147

[8] Vecchio F J, Collins M P. The Response of Reinforced Concrete to In-Plane Shear and Normal Stresses. Publication No. 82–03, Department of Civil Engineering, University of Toronto, Toronto, <month>March</month>1982, 332

[9] Vecchio F J, Collins M P, Aspiotis J. High-strength concrete elements subjected to shear. ACI Structural Journal, 1994, 91(4): 423–433

[10] André H M O. Toronto/Jajima study on scale effects in reinforced concrete element. Dissertation for the Doctoral Degree. Toronto: University of Toronto, Toronto, 1987, 267

[11] Khalifa J. Limit analysis and design of reinforced concrete shell elements. Dissertation for the Doctoral Degree. Toronto: University of Toronto, 1986, 314

[12] Kirschner U, Collins M P. Investigating the Behaviour of Reinforced Concrete Shell Elements. Publication No. 86–09, Department of Civil Engineering, University of Toronto, Toronto, 1986, 210

[13] Marti P, Meyboom J. Response of prestressed concrete elements to in-plane-shear forces. ACI Structural Journal, 1992, 89(5): 503–514

[14] Ohmori N, Takahashi T, Tsubota H, Inoue N, Kurihara K, Watanabe S. Experimental studies on non-linear behaviours of reinforced concrete panels subjected to in-plane shear. Journal of Structural and Construction Engineering, 1989, 403: 105–118 (in Japanese)

[15] Pang X, Hsu T T C. Constitutive Laws of Reinforced Concrete in Shear. Department of Civil and Environmental Engineering, University of Houston, Research Report UHCEE 92–1, Houston, 1992, 180

[16] Collins M P, Porasz A. Shear design for high-strength concrete. Comité Euro-International du Béton, CEB Bulletin d’Information,1989, 193: 77–83

[17] Zhang L, Hsu T T C. Behaviour and analysis of 100MPa concrete membrane elements. Journal of Structural Engineering, 1998, 124(1): 24–34

[18] Belarbi A, Hsu T T C. Constitutive laws of softened concrete in biaxial tension-compression. ACI Structural Journal, 1995, 92(5): 562–573

[19] Bentz E C, Vecchio F J, Collins M P. Simplified modified compression field theory for calculating shear strength of reinforced concrete elements. ACI Structural Journal, 2006, 103(4): 614–624

[20] Marti P. Basic tools of reinforced concrete beam design. ACI Structural Journal, 1985, 82(1): 46–56

[21] Schlaich J, Schäfer K. Towards a consistent design of structural concrete. PCI Journal, 1987, 32(3): 75–150

[22] Muttoni A, Schwartz J, Thürlimann B. Design of Concrete Structures with Stress Fields. Basel: Birkhäuser Verlag, 1997

[23] DIN EN 1992–1-1/NA:2013–04: Nationaler Anhang – National festgelegte Parameter - Eurocode 2: Bemessung und Konstruktion von Stahlbeton- und Spannbetontragwerken- Teil 1–1: Allgemeine Bemessungsregeln und Regeln für den Hochbau. DIN, European Standard, 2013

[24] Tizatto V, Shehata L C D. Longitudinal shear strength of wide compression flanges. Materials and Structures, 1990, 23(1): 26–34

[25] Sigrist V, Bentz E, Fernández Ruiz, M, Foster S, Muttoni A. Background to the Model Code 2010 Shear Provision – Part I: Beams and Slabs. Structural Concrete, 2013 14(3): 195–203

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg