Nonlinear analytical solution of a slender reinforced concrete element under axial tension

A new nonlinear analytical solution of the behavior of a slender reinforced concrete (RC) element under axial tension (known as "the tension stiffening problem") is developed. It refers to a concrete slender element with a central steel reinforcing bar (rebar) that is bonded to the concrete. The rebar's ends are loaded by tensile loads. An arbitrary nonlinear bond stress-slip relationship is considered, representing the nonlinear shear stress transmission at the rebar-concrete interface. The nonlinear solution is capable at solving the tension stiffening problem for any slip range. The solution provides the important parameters of the problem (e.g., slip at the RC element ends, rebar elongation, tensile stresses in concrete and rebar at the element center prior to cracking, and the cracking load levels). The first integral of the governing second-order nonlinear autonomous differential equation, accounting for the concrete-rebar interfacial slip, is derived, and using the boundary conditions, the slip at the RC element ends is obtained. Using the calculated slip, the other parameters are calculated. This solution procedure reduces the solution of a symmetric tension stiffening problem to sub-problems for uncracked sub-elements until a new crack is formed, after which the number of sub-elements doubles with half-length of each sub-element prior to crack formation. Since all sub-elements are identical, solution of a single sub-element is applied for all other sub-elements. The nonlinear solution is compared with experimental data and with approximate linear and bilinear solutions.