Rectangular flat finite element for modeling the process of crack formation

A rectangular flat finite element is proposed that allows modeling the process of crack formation without changing the initial elements' grid. The proposed finite element can be used to calculate structures made of reinforced concrete, masonry, fiber concrete, and other materials with low tensile strength. To calculate structures with existing cracks, their position can be specified as initial data. The finite element was formed based on the stress fields approximations and the principle of possible displacements to obtain the equilibrium equations of nodes. To calculate the stiffness matrix of the finite element, the principle of minimum additional energy was used, to which algebraic equilibrium equations were added using the Lagrange multiplier method. After a crack formation in the centre of finite element, additional degrees of freedom were introduced into its nodes, determining the possible mutual displacement of the element’s parts separated by the crack. The calculations were performed for a rectangular elastic bending beam with a low tensile strength. The reinforcement was located in the tensile zone of the beam. For comparison, the beam was also calculated using standard finite elements. The comparison of the results, including the crack width, for the two solutions showed that they coincided with high accuracy. The maximum displacements differ by 1.5%, the maximum stresses in compressed concrete and reinforcement differ by less than 1%. The crack width for the two solutions differs by no more than 5−7%.