1 Section for Structural Engineering, Department of Civil Engineering, Technical University of Denmark2 Department of Civil Engineering, Technical University of Denmark
Computational Modeling with Finite Elements for Lower Bound Limit Analysis of Reinforced Concrete Structures
For more than half a century, limit state analysis based on the extremum principles have been used to assess the load bearing capacity of reinforced concrete structures. Extensi- ve research within the field has lead to several techniques for performing such analysis manually. While these manual methods provide engineers with valuable tools for limit sta- te analysis, their application becomes difficult with increased structural complexity. The main challenge is to solve the optimization problem posed by the extremum principles. This thesis is a study of how numerical methods can be used to solve limit state analysis problems. The work focuses on determination of the load bearing capacity of reinforced concrete structures by employing the lower bound theorem and a finite element method using equilibrium elements is developed. The recent year’s development within the field of convex optimization is applied to solve the limit state problems. Three different element types have been developed and tested. The first is a solid tetra- hedral element with a linear stress distribution. The tri-axial stress state in the element is decomposed into concrete and reinforcement stresses, to which separate yield criteria are applied. The reinforcement is assumed to carry axial stresses only and is constrained by simple upper- and lower limits while the modified Coulomb criterion is applied to the concrete stresses. The element is verified by analytical solutions and used to model and analyze a console beam with complex reinforcement layout. The second element is a beam element capable of carrying loads in three dimensions. The element employs a zone model which provides a discrete representation of the in- ternal stress state in the beam. By applying the yield criterion on a stress state level, the element circumvents the need for a complex section force based yield criterion. The stresses are, similar to the solid model, decomposed into concrete and axial reinforcement stresses to which separate yield criteria are applied. An approximation to the modified Coulomb criterion using second-order cone constraints is developed for improved perfor- mance. An example is given in which an inverse T-beam is analyzed and the numerical results are compared to laboratory tests. The third and final element is a plane shell element capable of modeling membrane and plate bending behavior. The element employs a layered disk approach to create a discrete representation of the internal stresses. The stress state is separated and yield criteria are applied similar to the solid element. Because the transverse shear stresses are included in the modified Coulomb criterion, the element is capable of modeling the effects and combined section forces such as plate bending and transverse shear. Examples are given which illustrates how the element can model plate and disk structures and the importance of taking transverse shear into account for structural problems with combined bending and transverse shear is illustrated.