Engineering Structures 29 (2007) 899–903 www.elsevier.com/locate/engstruct
Behavior of beams with dowel action Bilal El-Ariss ∗ United Arab Emirates University, Civil and Environmental Engineering Department, P.O. Box 17555, Al Ain, United Arab Emirates Received 17 February 2004; received in revised form 25 June 2006; accepted 7 July 2006 Available online 6 September 2006
Abstract The phenomenon of dowel action as a shear transfer mechanism across cracks has long been recognized as an important component of the overall shear resistance capacity of reinforced concrete beams. In this paper, a simple analytical model for the dowel action of reinforcing bars crossing cracks is developed for analysis of reinforced concrete beams. This model is incorporated into a computer program that uses the displacement method and the initial stiffness procedure. The nonlinear behaviors of several reinforced concrete beams tested by others are analyzed. The beams are analyzed first with the dowel action neglected and then with the dowel action considered. It is found that in certain cases, the dowel action can have significant effects on the shear strength of reinforced concrete beams and that the theoretical results of the proposed model generally agree better with the experimental values when the dowel action is accounted for. c 2007 Published by Elsevier Ltd
Keywords: Reinforced concrete beams; Dowel force; Elastic foundation
1. Introduction Cracking in concrete beams may result in a significant reduction in their stiffness and strength. Although the flexural behavior of cracked reinforced concrete beams can generally be well predicted, accurate prediction of the shear behavior of reinforced concrete beams remains a formidable task due to the complexity of the shear transfer mechanism in the reinforced concrete. Shear resistance in reinforced concrete beams is provided by the shear transfer in the compression zone, aggregate interlock across the crack face, stirrups crossing the shear crack, and dowel action of longitudinal reinforcing bars crossing the crack in the concrete. The contributions of the compression zone, the stirrups, and the aggregate interlock are fairly well modeled in the literature, but so far the dowel action of the reinforcing bars has not been explicitly represented, despite the implicit belief in much of the current design thinking that dowel action is an important component of the shear resistance. The dowel action of reinforcing bars can play an important role if other contributions to shear transfer are relatively small as in the case of a beam with a small amount of web reinforcement or the case of a post-peak stage of the ∗ Tel.: +971 50 663 4601, +971 3 762 4601; fax: +971 3 762 3154.
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loading process. It may contribute significantly to the postpeak resistance and hence contribute to the shear ductility of concrete members. From a literature survey on finite element analysis of reinforced concrete structures covering papers published from 1985 to 1991 [1], it is noted that modeling of the dowel action has not been mentioned in any of the papers surveyed. This reflects to some extent the difficulties involved in modeling the behavior of the dowel action. There are some major difficulties in modeling the dowel action of reinforcement bars for finite element analysis. In experimental tests, the shear force transferred by the dowel action is quite difficult to measure because it is embedded with other shear transfer components. In fact, since the dowel action involves interaction between the reinforcement bars and the concrete near the cracks and the interaction stresses are extremely difficult to measure, many details of the dowel action have never been investigated. Consequently, experimental results on the dowel action have been rather limited. Even in finite element analysis, the mechanism of the dowel action is too complicated to describe. To analyze the details of the dowel action, the steel bars need to be individually modeled by finite elements and a very fine mesh has to be used for the concrete. As a result, the number of elements required would be very large. Furthermore, such individual modeling of the steel bars and concrete is not
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compatible with the common practice of modeling the concrete and the steel together in the analysis of reinforced concrete structures. To consider the dowel action in the global analysis of reinforced concrete structures, a simplified model of the dowel action that is compatible with the crack and reinforcement models is required. Thirdly, since the dowel action is usually more significant near peak load and at the post-peak stage, experimental testing or theoretical analysis extending into the post-peak range are needed to investigate the full effects of the dowel action, but such testing and analysis are generally quite difficult. To incorporate the effects of the dowel action in the nonlinear analysis of reinforced concrete beams, a simplified analytical model for the dowel action is described in this paper. The model is incorporated into a computer program that employs the displacement method and the initial stiffness procedure. 2. Analytical model The analytical model used to predict the behavior of a dowel bar embedded in concrete is based upon the work presented by Timoshenko and Lessels [2] for the analysis of beams on an elastic foundation. A beam on an elastic foundation is made of discrete springs that connect a beam to a rigid base. The dowel action behavior of the reinforcement bars crossing cracks in the concrete is analyzed by treating each reinforcement bar as a beam and the surrounding concrete as a bed of springs so that the reaction force of the foundation at any point may be assumed to be proportional to the deflection of the beam at that point. According to Timoshenko and Lessels, the differential equation for the deflection of a beam on elastic foundation is written as follows: EI
d4 y = −ky dx 4
(1)
where: k = stiffness of the elastic foundation (the concrete represents the flexible foundation); y = deflection. The solution to this differential equation is given by: y = eλx (A cos λx + B sin λx) + e−λx (C cos λx + D sin λx) s k λ= 4 4E s Is where: E s = modulus of elasticity of the steel bars;
(2) (3)
πd 4
Is = moment of inertia of the bar (equal to 64b in which db is the diameter of the bar); A, B, C, and D = constants determined from the boundary conditions for a particular problem. Cutting the reinforcing bar at the face of the crack, the bar may be treated as a semi-infinite beam resting on an elastic foundation and subjected to concentrated dowel force Vd and
Fig. 1. Semi-infinite beam on an elastic foundation.
Fig. 2. Forces acting on the dowel bar.
moment Mo applied at its end, as shown in Fig. 1. For a semiinfinite beam on an elastic foundation, the constants A and B are equal to zero and Eq. (1) becomes: y=
e−λx [Vd cos λx − λMo (cos λx − sin λx)] . 2λ3 E s Is
(4)
Differentiating Eq. (4) with respect to x gives the slope, dy/dx: e−λx dy = 2 [(2λMo − Vd ) cos λx − Vd sin λx] . dx 2λ E s Is
(5)
Applying the solution for a semi-infinite beam on an elastic foundation to dowel bars crossing a joint in a concrete pavement, Friberg’s [3] developed equations for the slope and deflection of a dowel at the face of the joint are used to determine the slope and deflection of a dowel at the face of a crack in concrete. In Friberg’s analysis, the stiffness of the elastic foundation, k, was replaced with the expression K o b. The modulus of the dowel support, K o , denotes the reaction when the deflection is equal to unity and b represents the dowel bar diameter. Friberg’s developed equations were derived assuming a dowel bar of semi-infinite length. However, many engineers view Friberg’s work as the authoritative analysis on the behavior of dowel bars to date. Therefore, Friberg’s equations were used in accomplishing the theoretical work associated with this research. Assuming that an inflection point exists in the dowel at the center of the crack, the forces acting on the portion of the dowel within the crack width, z, are shown in Fig. 2. Substituting −( V2d z ) for Mo and setting x equal to zero, Eqs. (4) and (5) become Eqs. (6) and (7) for the slope and deflection of the dowel at the face of a crack in concrete, as shown in Fig. 3: dyo Vd = 2 (1 + λz) dx 2λ E s Is Vd yo = 3 (2 + λz) . 4λ E s Is
(6) (7)
The stiffness of the elastic foundation (concrete surrounding the dowel bars) is an important parameter in the equations
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the center of the crack to the level of the bar in the concrete; therefore: dyo z ∆ = 2 yo + dx 2 Vd (2 + λz) + (1 + λz) (z) . (11) = λ 2λ2 E s Is
Fig. 3. Slope and deflection of dowel at the face of the crack.
presented in this paper. Before these equations can be used, a value for the elastic foundation stiffness is needed. For the elastic foundation stiffness of the surrounding concrete, k, the following data-fitting expression proposed by Soroushian et al. [4] is used: p 127c1 f c0 (8) k= 2/3 db where: f c0 = compressive strength of the concrete in N/mm2 ; db = diameter of the bar in mm; c1 = coefficient ranging from 0.6 for a clear bar spacing of 25 mm to 1.0 for larger bar spacing. The load–deflection response for dowel bars embedded in concrete proposed by Millard and Johnson [5] is adopted in this research. Although it has been suggested that dowel strength across a shear plane is owing to a combination of direct shear, kinking and flexure of the reinforcing bars, Millard and Johnson have illustrated that flexure of the bars predominates, since there is a significant amount of deformation in the underlying concrete cover. They proposed the following load–deflection response for dowel bars embedded in concrete: −k∆ (9) Vd = Vu 1 − exp Vu where Vd = dowel force at a shear displacement ∆ at a crack; Vu = the ultimate dowel force. When the dowel deformation is not too large and none of the materials have yielded, the dowel force–displacement relation is linearly elastic. However, when the elastic limit is exceeded, the dowel action becomes plastic. At the ultimate limit state, local crushing of the surrounding concrete and/or yielding of the dowel bar occurs. Based on experimental results, Dulacska [6] has given the following equation for estimating the dowel force at ultimate limit state Vu : q Vu = 1.27db2 f c0 f y (10) where: f y = yielding strength of the dowel bar. The dowel displacement ∆ used in Eq. (9) can be assumed equal to the distance from the inflection point in the dowel at
Substituting Eqs. (10) and (11) in Eq. (9) will yield the dowel force of the reinforcing bar: q Vd = 1.27db2 f c0 f y −kVd (2+λz) + + λz) (1 (z) λ q . (12) × 1 − exp 2.54λ2 E s Is db2 f c0 f y u
The forces acting on the portion of the dowel within the crack width, z, are as shown in Fig. 2 where Mo is equal to −( V2d z ). The moment produced by the dowel force, Vd , will tend to reduce the moment applied at the section where the crack intersects the reinforcing bar. The resultant moment is used in the standard section analysis to compute the strain and curvature. In the standard section analysis, the resultant moment is applied at its corresponding beam cross section. Due to the application of the moment a change in the strains and stresses will occur at the section. Two parameters, strain and stress, are used to define the strain and stress distributions. These two parameters are then obtained from the equilibrium requirements. The analysis is repeated for a number of sections. An arbitrary number of sections along the beam is chosen and incorporated in the computer program to perform the analysis. When the dowel action of the reinforcing bars is not considered, the standard section analysis is performed using the moment applied at the section and not the resultant moment above. It is worth noting that the section analysis employed in the computer program along with the displacement method and the initial stiffness procedure has an advantage over the standard finite element method. The essential feature of the analysis is that the actual deflected shape is obtained by integrating the actual strains and curvatures. In the finite element method, the deflected shape of a member is usually assumed as a function of the displacements at the nodes and equilibrium between the external and internal forces is satisfied only at the nodes. 3. Verification of the analytical model To verify the reliability of the proposed analytical model, a comparison with experimental and analytical work conducted by other researchers is carried out. The deep beams tested by Ashour [7] are analyzed and the analytical results are compared to the experimental results. The beams are analyzed twice, first with the dowel action neglected and then again with the dowel action incorporated, in order to study the significance of the dowel action of the main reinforcement bars contained in these beams. Beams, CDB1, CDB2 and CDB3, are selected for the analysis. The details of the beams are shown in Fig. 4. The top and bottom longitudinal reinforcement bars have yielding
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(a) Beam CDB1.
(b) Beam CDB2.
(c) Beam CDB3. Fig. 4. Details of the beams analyzed (dimensions in mm).
strengths of 500 MPa and 400 MPa, respectively. The web reinforcement was 8 mm diameter steel bars with yielding strength of 370 MPa. The compressive strengths of the concrete for the beams CDB1, CDB2 and CDB3 were 30.0 MPa, 33.1 MPa and 22.0 MPa, respectively. The load–deflection curves are plotted in Fig. 5. It is seen that when the dowel action is not taken into account, the predicted strengths are lower than the corresponding experimental values. However, when the dowel action is taken into account, the load–deflection responses of the beams are in better agreement with the test results. This reveals that the contribution of the dowel action has a significant effect on the behavior of the beams. The effect of the dowel action becomes evident when the applied load approaches the peak. Beyond the peak, the effect of the dowel action is even more significant especially when the aggregate interlock action along the cracks drops due to gradual increase of crack widths. The importance of the dowel action increases as the amount of web reinforcement decreases. In beam CDB1, almost all of the vertical shears are resisted by the web reinforcement and thus the contribution of the dowel action is relatively small. In beam CDB3, there is no web reinforcement and consequently the dowel action plays a more important role in resisting the applied shear force, which agrees with He and Kwan [8]. They modeled the dowel action of reinforcement bars for finite element analysis of concrete structures and showed that the dowel action can have significant effects on the shear strength and ductility of reinforced concrete beams.
Fig. 5. Load–deflection curves (DA is dowel action).
4. Conclusions A simple analytical model for the dowel action of reinforcing bars crossing cracks in concrete is developed and incorporated into a computer program for the nonlinear analysis of reinforced concrete beams. The behavior of the dowel bar is derived based on the beam on an elastic foundation theory. Application of the dowel action model to the analysis of deep reinforced concrete beams tested by others verified that the proposed dowel action model can be used to predict the behavior of shear critical reinforced concrete members. The analytical results also showed that the dowel action could have significant effects on the behavior and ductility of the reinforced concrete beams especially when the amount of web reinforcement in the beam is small. Therefore, in the nonlinear analysis of shear critical reinforced concrete members, the dowel action should be taken into account. It is recommended to carry out a parametric study to investigate the influence of some parameters such as bar diameters and amount of reinforcement on the dowel action as a shear transfer mechanism across cracks. Comparison with other experimental results would provide more complete and satisfactory results.
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Further research on the bearing capacity of dowel action in concrete structures reinforced with fiber reinforced plastic (FRP) bars would provide some insight on the importance of the dowel action. References [1] Darwin D, editor. Finite element analysis of reinforced concrete structures II. New York: ASCE; 1993. p. 203–32. [2] Timoshenko S, Lessels JM. Applied elasticity. Pennsylvania: Westinghouse Technical Night School Press; 1925. [3] Friberg BF. Design of dowels in transverse joints of concrete pavements.
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Transactions, American Society of Civil Engineers 1940;105(2081). [4] Soroushian P, Obaseki K, Rajos MC. Bearing strength and stiffness of concrete under reinforcing bars. ACI Materials Journal 1987;84(3):179–84. [5] Millard SG, Johnson RP. Shear transfer across cracks in reinforced concrete due to aggregate interlock and dowel action. Magazine of Concrete Research 1984;36(126):9–21. [6] Dulascka H. Dowel action of reinforcement crossing cracks in concrete. ACI Structural Journal 1972;69(12):754–7. [7] Ashour AF. Tests of reinforced concrete continuous deep beams. ACI Structural Journal 1997;94(1):3–12. [8] He XG, Kwan AKJ. Modeling dowel action of reinforcement bars for finite element analysis of concrete structures. Computer & Structures 2001;79: 595–604.
