Definition

Incremental formulation is a method used in numerical analysis, particularly in finite element analysis (FEA), to solve problems involving nonlinear behavior by applying loads or displacements in small increments. This approach allows the analysis to account for changes in material properties, geometry, or boundary conditions that occur as the load is applied. Incremental formulation is essential for accurately modeling complex behaviors such as plasticity, large deformations, and contact problems.

Principle

In incremental formulation, the total response of a structure is determined by summing up the responses to a series of small load increments. At each increment, the stiffness matrix and other parameters are updated to reflect the current state of the structure, allowing the analysis to capture nonlinearities. The process involves:

• Incremental Load Application: The total load is divided into smaller increments, and the structure’s response is calculated for each increment.
• Equilibrium Check: After applying each increment, the equilibrium of the structure is checked, and any necessary adjustments are made to account for nonlinear behavior.
• Iteration: The process is iterative, with the analysis proceeding incrementally until the total load is applied and the final state of the structure is determined.

Context of Use

Incremental formulation is critical in various applications within geotechnical and structural engineering, particularly when dealing with nonlinear behavior:

• Plasticity: In materials that exhibit plastic deformation, incremental formulation is used to model the transition from elastic to plastic behavior and to predict the permanent deformations.
• Large Deformations: For structures undergoing large deformations, such as flexible membranes or cables, incremental formulation helps accurately capture the changes in geometry and the resulting stresses.
• Contact Problems: In contact analysis, where two or more bodies interact, incremental formulation allows the analysis to account for the evolving contact conditions as the load is applied.
• Nonlinear Material Behavior: Materials like soil and rock, which exhibit nonlinear stress-strain relationships, are analyzed using incremental formulation to ensure accurate predictions of their behavior under load.

Advantages

• Accurate Modeling of Nonlinear Behavior: Incremental formulation provides a robust method for capturing complex, nonlinear responses in structures and materials, ensuring accurate analysis results.
• Flexibility in Analysis: This approach allows for the incorporation of varying material properties, boundary conditions, and loading scenarios, making it adaptable to a wide range of engineering problems.

Limitations

• Increased Computational Effort: Incremental formulation requires more computational resources and time, as the analysis must be repeated for each load increment.
• Convergence Challenges: In highly nonlinear problems, convergence may be difficult to achieve, requiring careful selection of load increments and iterative methods to ensure accurate results.

Summary

Incremental formulation is a key method in numerical analysis for solving problems involving nonlinear behavior by applying loads or displacements incrementally. This approach allows for the accurate modeling of complex material behaviors, large deformations, and contact interactions, making it essential in geotechnical and structural engineering. While it increases computational effort, the ability to capture nonlinearities ensures that incremental formulation provides reliable and detailed analysis results.