- Open Access
Shear Resistant Mechanism into Base Components: Beam Action and Arch Action in Shear-Critical RC Members
- Je-Pyong Jeong^{1}Email author and
- Woo Kim^{2}
https://doi.org/10.1007/s40069-013-0064-x
© The Author(s) 2014
- Received: 27 February 2013
- Accepted: 13 December 2013
- Published: 7 March 2014
Abstract
In the present paper, a behavioral model is proposed for study of the individual contributions to shear capacity in shear-critical reinforced concrete members. On the basis of the relationship between shear and bending moment (V = dM/dx) in beams subjected to combined shear and moment loads, the shear resistant mechanism is explicitly decoupled into the base components—beam action and arch action. Then the overall behavior of a beam is explained in terms of the combination of these two base components. The gross compatibility condition between the deformations associated with the two actions is formulated utilizing the truss idealization together with some approximations. From this compatibility condition, the ratio of the shear contribution by the tied arch action is determined. The performance of the model is examined by a comparison with the experimental data in literatures. The results show that the proposed model can explain beam shear behavior in consistent way with clear physical significance.
Keywords
- arch action
- beams
- reinforced concrete
- truss model
- shear strength
1 Introduction
Up to now, the beam shear problem remains one of the more controversial aspects of structural reinforced concrete analysis and design, and it has been generally agreed that the truss model theory provides a more promising way to treat the problem. That is not only because it provides a clear concept of how a reinforced concrete beam resists shear after cracking, but also because the effect of various loading conditions can be included in a logical way (ASCE-ACI Committee 445 1998; ASCE-ACI Committee 426 1973; Hsu 1993).
The classical shear analysis of Ritter and Mörsch explains the shear behavior in the cracked state by a truss analogy, using a truss with parallel chords with 45° inclined concrete struts and no stresses across the cracks. In this model the bending moment is carried by the top chord (concrete compression zone) and the bottom chord (main longitudinal reinforcement), and the applied shear force is fully carried by the web by means of inclined compressive stresses in the concrete and tension in the stirrups. This simplified version of truss model has long provided the basis for the formulation of general shear design codes, such as in ACI-318 (1999) and Eurocode-2 (Commission of the European Communities 1991). During last four decades, the concept of the truss model theory has been greatly extended, and now several approaches have been developed (Marti 1985; Nielsen 1984; Vecchio and Collins 1986; Schlaich et al. 1987; Ramirez and Breen 1991). An excellent review of the current theories as well as available experimental evidences are given by ASCE-ACI Committee (ASCE-ACI Committee 445; ASCE-ACI Committee 426).
One approach (ACI Committee 318) has been to add a concrete contribution term to the web shear reinforcement contribution, assuming a parallel chord truss with the strut angle of 45°. Another approach (CEB/FIP 1990; Commission of the European Communities 1991) has been the use of a parallel chord truss with a variable angle of inclination of the diagonal struts. This approach is referred to as the standard truss model with no explicit concrete contribution, and is explained by the existence of aggregate interlocking and the dowel action, which make a lower inclination of the concrete struts and a higher effectiveness of the stirrups. A combination of the variable-angle truss with parallel chords and a concrete contribution has also been proposed. This approach has been referred to as the modified truss model (American Association of State Highway and Transportation Officials 2002; Ramirez and Breen 1991).
Resistant components of various truss models
Model | Resistant components | Responses | |||||
---|---|---|---|---|---|---|---|
V _{c} | V_{ s }(=V_{ t }) | θ | f_{ v } stirrup stress | ΔT tension shift | |||
V_{uc} (= v_{a)} | V_{ci} | Vd | |||||
Classical truss model | X | X | X | ○ | 45° | \( \frac{1}{{\rho_{v} b_{w} z_{o} }}(V) \) | 1/2(V) |
Standard truss model type (a) | X | X | X | ○ | Variable | \( \frac{\tan \theta }{{\rho_{v} b_{w} z_{o} }}(V) \) | 1/2(V) cotθ |
Modified truss model | |||||||
ACI | ○ Lump sum V_{ c } (fixed) | ○ | 45° | \( \frac{1}{{\rho_{v} b_{w} z_{o} }}(V - V_{c} ) \) | 1/2(V − V_{ c }) | ||
EC-2 | ○ Lump sum V_{ c } (fixed) | ○ | Variable | \( \frac{\tan \theta }{{\rho_{v} b_{w} z_{o} }}(V - V_{c} ) \) | 1/2(V − V_{ c }) cotθ | ||
AASHTO LRFD | X | ○ (vari able) | X | ○ | Variable | \( \frac{\tan \theta }{{\rho_{v} b_{w} z_{o} }}(V - V_{ci} ) \) | 1/2(V − V_{ ci }) cotθ |
Complex truss model | |||||||
Type (b) | ○ | X or ○ | X | ○ | Irregular variable | Undefined | |
Type (c) | ○ | X or ○ | X | ○ | Irregular variable | Undefined | |
Type (d) | ○ | X or ○ | X | ○ | Irregular variable | Undefined |
Therefore, the present work is intended to numerically formulate the truss model with an inclined compression chord shown in Fig. 1 by decoupling the beam behavior into the tied arch and the beam. The theoretical base concept for the present approach is based on the relationship between shear and bending moment in a cracked reinforced concrete beam, i.e., V = dM/dx. Utilizing some idealizations together with the recent elaborations by Collins and Mitchell (1991) and Hsu (1993), a gross compatibility condition is established and formulated between the deformation associated with the tied arch and the deformation with the web. Thereby, the beam shear resistant mechanism is decoupled into the base components. The performance of the present approach is briefly examined by a comparison with the existing experimental data and a sensitivity study. It is also shown that the theoretical results can explain in a rigorous and consistent way the experimentally observed behavior of beam failing in shear.
2 Derivation of Base Concept
As known from various proceeding studies, the first term arises from the transmission of a steel force into the concrete by means of bond stresses, and it is said to be the shear resistant component by beam action (Kani 1964; Park and Paulay 1975). Consider a segment cut out from the beam between two adjacent vertical cross sections distance dx apart in Fig. 2a, the difference of tension dT causes shear force on the bottom face of the web element mnop as shown in Fig. 2b. As the same manner, the difference of compressive resultant dC acts on the upper face. These shear forces on the top and bottom of the web element produce a couple moment zdT (=zdC), which must be balanced by the moment of shear forces acting on the vertical faces mn and op. Thus, the vertical shear force is expressed by the first term of Eq. (1).
The second term in Eq. (1) directly implies the vertical component of the inclined compression resultant force C, and it is referred to the shear component by arch action (Kani 1964); Park and Paulay 1975). That is because the compression resultant is equal to the tension resultant T and the slope of the resultant is mathematically expressed dz/dx as shown in Fig. 2c. This means that the beam behaves as a tied-arch, and a part of the applied shear is carried by the inclined top chord.
2.1 Smeared Truss Idealization with Inclined Chord
In a simple strut-and-tie model, the tensile force of the tie (bottom chord) is constant throughout the span (dT/dx = 0), so that beam action cannot be developed in the web shear element. Accordingly, the shear and moment must be fully resisted by the inclined top (strut) and bottom chords, as shown in Fig. 3c. From the view of the present model, it is said that a simple strut-and-tie model is the extreme case of the present model when α is 1.0. On the other hand, a parallel chord truss model (dz/dx = 0 as seen in Fig. 3b) cannot rely on the arch action to sustain shear, thus it is the other extreme case of the present model with α of 0. It may be realized therefore that the internal force flow of usual beams can be closely described by a proper assignment of the value of α, as conceptually illustrated in Fig. 3d.
2.2 Gross Compatibility Condition
The idealization above make it possible to evaluate the state of the cross sectional deformation of a reinforced concrete beam. Consider the deformation of the element mnop in a beam under a concentrated load as shown in Fig. 4a. The final deformation of the element can be decomposed into the two base components—the deformation associated with the arch action and the deformation associated with the beam action. As stated before, the bending moment causes the internal couple C and T at the chords, resulting in the axial shortening of the top chord and the axial elongation of the bottom chord. Thus these axial deformations eventually produce a bending curvature on the element as shown in Fig. 4b.
The element further undergoes a shear deformation because it is subjected to a pure shear dT/dx, which is equal to (1 − α)V/z, as shown in Fig. 4c. Linear distortion can be assumed with the average shear strain of γ_{ w }. It is seen from the figure that the shear strain is equal to the amount that the upper edge displaces horizontally with respect to the lower side divided by the depth of the element. Consequently, this shear deformation of the web shear element should be compatible with the relative displacements of the top and bottom chord with respect to the bending planes m′–n′, which are designated by u_{ m } and u_{ n } respectively in Fig. 4c, because the element is connected to the chords. The relative displacements of the chords can be visualized easily if the bending deformation and the couple C and T acting on the chords are omitted and the all elements are stretched as shown in Fig. 4e.
From the view of the gross compatibility relationship above, it can be seen that the ratio of contribution by each action to total shear resistance in a beam is dependent on the relative stiffness ratio between the chords and the web (Kim and Jeong 2011a, 2011b, 2011c).
2.3 Simplified Arch Shape Function
3 Formulation
3.1 Web Shear Element
Substituting Eqs. (13), (14) and (15) for ɛ_{1}, ɛ_{2} and ɛ_{ t } respectively in Eq. (16), the shear strain γ_{ w } of the web element is eventually expressed in terms of f_{1}, f_{2} and θ.
3.2 Redistribution of Resultants in a Section
3.3 Relative Displacements of Top and Bottom Chord
3.4 Solution Algorithm
Then, check that γ_{ w }from Eq. (16a) = γ_{ w } from Eq. (16b) or not.
If γ_{ w }from Eq. (16a) ≠ γ_{ w } from Eq. (16b), return to Step-2.
If γ_{ w } ≠ (u_{ m } + u_{ n })/z, return to Step-1 and repeat
If γ_{ w } ≅ (u_{ m } + u_{ n })/z, take the last assumed α value.
With the help of a spread sheet calculator, the procedure above is easily performed. As a tip for fast iteration, it is recommended to use the inverse value of the shear span-to-depth ratio of the beam considered as an initial value of α in Step-1.
4 Verification
Comparisons with, the test results performed by Leonhardt (1965)
Specimen |
ET1 |
ET2 |
ET3 |
ET4 |
---|---|---|---|---|
Beam properties | ||||
f_{ ck } (MPa) | 27.93 | 27.93 | 27.93 | 27.93 |
b_{ w } (cm) | 30 | 15 | 10 | 5 |
d (cm) | 30 | 30 | 30 | 30 |
a (cm) | 105 | 105 | 105 | 105 |
a/d | 3.5 | 3.5 | 3.5 | 3.5 |
ρ (%) | 1.40 | 2.80 | 4.20 | 8.40 |
ρ_{ v } (%) | 0.17 | 0.34 | 0.51 | 1.03 |
f_{ y } (MPa) | 460 | 460 | 460 | 460 |
f_{αv} (MPa) | 314 | 314 | 314 | 314 |
V_{ n,f } (kN) | 140.9 | 140.9 | 140.9 | 140.9 |
Measured value | ||||
V_{ u } (kN) | 142.2 | 116.7 | 98.1 | 88.3 |
f_{ v } at V_{ u } (MPa) | 161.7 | 314 | 314 | 314 |
Predicted value | ||||
α_{ u } | 0.360 | 0.395 | 0.415 | 0.462 |
θ_{ u } (x = a/2) (°) | 44.6 | 45.4 | 43.6 | 41.4 |
z_{ x } (x = a/2) (cm) | 21.04 | 20.46 | 20.18 | 19.63 |
f_{ v } at V_{ u } (MPa) | 150.3 | 315.7 | 294.2 | 284.4 |
V_{ α } = α_{ u }V_{ u } | 51.2 | 46.6 | 42.8 | 41.2 |
V_{ ci }by Eq. (26) | 74.5 | 37.3 | 24.8 | 12.4 |
V_{ s } (kN) | 16.5 | 32.6 | 34.2 | 35.9 |
V_{ u } = V_{ α } + V_{ ci } + V_{ s } (kN) | 142.2 | 116.5 | 101.8 | 89.5 |
V_{ u Predicted } / V_{ u Measured } | 1.00 | 0.99 | 1.04 | 1.01 |
Failure mode | Flexure | Stirrups yielding |
The α_{ u } of the four beams is also listed in Table 2. And it is seen that as the web width decreases, the higher value of α_{ u } is resulted in with varying between 0.29 for ET1 and 0.46 for ET4. This means that 29–46 % of the applied shear force is carried by the arch action. It may be worthwhile to note that Leonhardt stated in his paper (1965) that 15–25 % of total shear was carried by the inclined compression chord in those beams.
Figure 13 indicates that the a/d ratio is the most dominant parameter that affects magnitude of α, and the value decreases with increasing a/d ratio. The rate of decrease is larger for high ratio of web reinforcement. Figure 13 also shows that the longitudinal steel ratio ρ has also a pronounced influence on the variation of α, and the higher ρ causes the greater α. Such trend is well agreed with the discussion that is based on the relative stiffness ratio between the chords and the web element.
4.1 Section Analysis
In determining the geometry of the present approach above, average stresses and average strains have been considered. However, the local stresses that occur at a crack are different from the calculated average values. At a crack the concrete tensile stress goes to zero, whilst the steel tension becomes larger. Also, the shear behavior of a beam is mainly governed by the forces transmitted across the crack.
The sum of the first two terms is normally referred to the concrete contribution V_{ c }, and the last term is referred to the steel contribution. In view of the present approach, the first term is the component resisted by the arch action, and the sum of the last two terms, which is equal to (1 − α)V, is the component resisted by the web element by means of the beam action.
4.2 Steel Stresses
The accuracy of the theoretically calculated tension of the vertical tie (stirrup) and the horizontal tie (longitudinal steel) may strongly confirm the rationale of the present approach although the solution procedure is too complicated to be used. The fact that the simple practical calculation method is accurate enough may also confirm the practical applicability of the present approach. Moreover, it can be seen from Fig. 16 that the shear prior to diagonal cracking is maintained during the stirrup stresses rise to yield level. This confirms the fact that both the arch and the beam action are the essential mechanisms in resisting the applied load in stabilized stage up to failure.
4.3 Ultimate Shear Strength V_{ u }
5 Conclusions
On the basis of the relationship between shear and the rate of change in bending moment (V = dM/dx = zdT/dx + Tdz/dx) in reinforced concrete beams subjected to shear and bending, a behavioral model has been proposed in the present paper. In the model the rate of the change in the lever arm (dz/dx) is accounted for, so that the shear resistant mechanism has been decoupled into two base components—the arch action and the beam action. The ratio (denoted by factor-α) of contribution to shear resistance by the tied arch action in a beam is numerically derived from the gross compatibility of deformations associated with the base actions. Then, the actual behavior of shear-critical beams is formulated by means of interpolating between the sectional approach and the tied arch approach using the value of the factor-α. The adequacy of the new approach has been briefly examined by some test results in literatures, and the results show an excellent agreement between the predicted and the measured. From the present study, it can be concluded that the factor-α is appeared to be the most crucial parameter for understanding the behavior of shear-critical reinforced concrete members.
Declarations
Acknowledgments
This work was supported by LINC (Leaders in Industry-university Cooperation) in Honam University and NRF (National Research Foundation of Korea). The authors wish to gratefully acknowledge this financial support.
This article is published under license to BioMed Central Ltd. Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Authors’ Affiliations
References
- ACI Committee 318. (1999). Building code requirement for reinforced concrete and commentary (318R-99) (p. 391). Detroit, MI: ACI.Google Scholar
- American Association of State Highway and Transportation Officials (2002), AASHTO LRFD Bridge Design Specifications, 2002 Interim Revisions, pp. 305–315.Google Scholar
- ASCE-ACI Committee 426. (1973). The shear strength of reinforced concrete members. Journal of Structural Division, ASCE,99(6), 1091–1187.Google Scholar
- ASCE-ACI Committee 445. (1998). Recent approaches to shear design of structural concrete. Journal of Structural Engineering,124(5), 1375–1417.View ArticleGoogle Scholar
- Bhide, S. B., & Collins, M. P. (1989). Influence of axial tension on the shear capacity of reinforced concrete member. ACI Structural Journal,86(5), 570–581.Google Scholar
- Collins, M. P., & Mitchell, D. (1991). Prestressed concrete structures (pp. 210–220). Eaglewood Cliffs, NJ: Prentice-Hall.Google Scholar
- Comite Euro International Du Beton (CEB/FIP)(1990), CEB-FIP Model Code for Concrete Structures, Bulletin d’Information No. 124/125, p. 437.Google Scholar
- Commission of the European Communities (1991), Eurocode No. 2: Design of Concrete Structures, Part 1: General rules and Rules for Buildings, ENV 1992-1-1, p. 253.Google Scholar
- Hsu, T. T. C. (1993). Unified theory of reinforced concrete (pp. 250–350). Boca Raton, FL: CRC.Google Scholar
- Kani, G. N. J. (1964). The riddle of shear failure and its solution. ACI Journal,61(4), 441–467.Google Scholar
- Kim, W., & Jeong, J.-P. (2011a). Non-Bernoulli-compatibility truss model for RC member subjected to combined action of flexure and shear, part I-its derivation of theoretical concept. KSCE Journal of Civil Engineering,15(1), 101–108.View ArticleGoogle Scholar
- Kim, W., & Jeong, J.-P. (2011b). Non-Bernoulli-Compatibility truss model for RC member subjected to combined action of flexure and shear, part II-its practical solution. KSCE Journal of Civil Engineering,15(1), 109–117.View ArticleGoogle Scholar
- Kim, W., & Jeong, J.-P. (2011c). Decoupling of arch action in shear-critical reinforced concrete beam. ACI Structural Journal,108(4), 395–404.Google Scholar
- Kim, D.-J., Kim, W., & White, R. N. (1998). Prediction of reinforcement tension produced by arch action in RC beams. ASCE, Journal of Structural Engineering,124(6), 611–622.View ArticleGoogle Scholar
- Leonhardt, F. (1965). Reducing the shear reinforcement in reinforced concrete beams and slabs. Magazine of Concrete Research,17(53), 187–198.View ArticleGoogle Scholar
- Lorentsen, M. (1965). Theory for the combined action of bending moment and shear in reinforced concrete and prestressed concrete beams. ACI Journal,62(4), 420–430.Google Scholar
- Marti, P. (1985). Basic tools of reinforced concrete beam design. ACI Journal,82(1), 46–56.MathSciNetGoogle Scholar
- Nielsen, M. P. (1984). Limit analysis and Concrete Plasticity. Eaglewood Cliffs, NJ: Prentice-Hall. 420.Google Scholar
- Park, R., & Paulay, T. (1975). Reinforced concrete structures (pp. 133–138). New York, NY: Wiley.View ArticleGoogle Scholar
- Ramirez, J. A., & Breen, J. A. (1991). Evaluation of a modified truss model approach for beams in shear. ACI Structural Journal,88(5), 562–571.Google Scholar
- Schlaich, J., Schafer, I., & Jennewein, M. (1987). Towards a consistent design of structural concrete. PCI Journal,32(3), 74–150.View ArticleGoogle Scholar
- Taylor, H. P. J. (1974). The fundamental behavior of reinforced concrete beams in bending and shear (pp. 43–77). Detroit, MI: ACI SP-42.Google Scholar
- Vecchio, F. J., & Collins, M. P. (1986). The modified compression field theory for reinforced concrete elements subjected to shear. ACI Journal,83(2), 219–231.Google Scholar