Nonlinear analytical solution for the pullout of a steel rebar from a reinforced concrete element
摘要
Experimental and theoretical investigation of the pullout problem is a fundamental and important issue in structural concrete. The pullout test focuses on the interfacial bond properties which are essential for complete modelling of the composite behavior of reinforced concrete. Using the experimentally based bond stress-slip relationship in numerical analysis, the state of stress and strain along an embedded rebar in concrete man be calculated. This is important for determination of the states of stress and strain and for assessing the cracking and crack widths in real structural elements. Thus, it is essential for determining how well a concrete structure will perform under stress in real-world applications and help in optimizing structural design and preventing potential failures. Since the experimental bond stress-slip relationship is highly nonlinear, a close representation of its effect on the reinforced concrete element behavior requires an appropriate nonlinear model. This has been the motivation for the present new nonlinear analytical solution for the pullout problem that is presented in this paper, aiming to determine the complete behavior and simulate typical tests. A new nonlinear analytical solution to the pullout problem has been developed in the present article. It refers to a steel reinforcing bar (rebar) of short length that is centrally embedded in a concrete element A tensile pull-out load is applied to the rebar outer end whereas the other embedded end is stress free. An experimentally based nonlinear bond stress-slip relationship is considered, representing the nonlinear shear stress transmission at the rebar-concrete interface. A new nonlinear solution to the problem is developed for the entire pullout problem. The analytical solution is developed for two loading protocols: force control and displacement control. The novelty of the developed approach is that for a given arbitrary nonlinear bond stress-slip relationship an analytical method is presented, based on a second-order nonlinear autonomous differential equation that describes the behavior of the concrete-rebar interface slip. The first integral is written, and then, using boundary conditions, the slips at the loaded and free ends of the RC element are obtained. These are the most important parameters of the pullout problem. If the main interest is not on the variations of stresses, strains, and displacements along the element, but is mainly focused on the maximum values of these parameters, then this stage completes the solution of the problem. Otherwise using the magnitude of the slips at the loaded and free ends, the above distributions can be calculated, and a detailed solution is obtained. The nonlinear solution is compared to experimental data and good predictions of the proposed approach with the measured data is demonstrated. The above solutions were obtained for cases of force control test and displacement control test. It has been found that force control is possible if the tensile force does not exceed a certain critical value. This critical force maintains stable equilibrium with the bond-stress distribution along the rebar as long there exists a segment along the rebar that the bond stresses may increase upon further loading. For any larger slip the total pullout force that is required to maintain equilibrium is decreasing and therefore a force control protocol is impossible. In contrast, the displacement (slip) of the loaded end of the rebar is continuously increasing, hence the entire pullout history may be well controlled by the displacement control protocol.