Non-linear P-delta analysis

In cases where the geometric deformation of the structure is considered to be significant, account of the deformation of the model and the subsequent impact on the stiffness of the analytic model can be considered by the use is the P-delta analysis.

Within MasterFrame, two types of P-delta analysis are available. Both of these methods use an iterative approach to modify the model geometry under load and so modify the stiffness matrix of the analytic model.

Geometric Stiffness

The Geometric Stiffness method is a two-pass iteration method. The first iteration determines the axial forces in the analytic members. The stiffness matrix is then modified to account for the axial force, where compressive force lead to a reduction in the bending stiffness of the member and tensile force produces an increase in the stiffness of the member in bending. The modified bending stiffness terms are calculated using stability functions, which depend on the axial force in the member. The second iteration is then carried out using this modified stiffness matrix.

The Geometric Stiffness method accounts for stress stiffening in the analysis and so is a P-δ method.

The Geometric Stiffness method is applicable where the lateral deflections of the structure, which are not taken into account in the modification of the stiffness method, are significant. Where lateral deformations of the structure are considered to be significant, a full non-linear iterative approach is required.

Newton-Raphson method

The Newton-Raphson method is a multi-pass iterative method which accounts for vertical and lateral deflections in the frame, accounting for both geometric deformation of the structure as a whole, and also the effect of the structure deformation on the stress in individual analytic members. As such, the Newton-Raphson method accounts for the P-Δ effects on the overall structure, while some P-δ effects are considered depending on the arrangement of the nodes within a MasterFrame physical member. 

At each stage of the iteration process, the deformation of the structure is determined from the analysis and the stiffness matrix is then modified to account for these deformations in the next iteration. At each stage, the load on the structure can be either the full load of defined on the structure, or an incremental approach can be used, where the load is increased in a number of increments, where the number of increments can be defined by the user.

The P-δ effects include the deformation of the individual members, that is, the deformation of the member along its length. However, the deformation of model is determined at the analytical nodes within the model. Therefore, to capture the deformation of a member, the member requires intermediate analytic nodes. Therefore, in a model where the MasterFrame physical members have no intermediate nodes, the in-member deformations are not accounted for in the analysis and in this case the analysis is purely a P-Δ analysis. To take account of the P-δ effects on any particular member, it is necessary to include nodes along the member length. However, it must be noted that each node adds 6 degrees of freedom the model, so introducing additional nodes will increase the number of equations to be solved and so increase the size of the stiffness matrix. Introducing a large number of analytic nodes can lead to an increase in the time to analyse each iteration of each load case, leading to longer analysis times.

At each iteration of the Newton-Raphson analysis, the convergence of the analysis is measured against a convergence criterion. This measures the change in the solution for each iteration and determine whether or not the method will converge to a solution. Where the process does not converge, this suggests that the structure is unstable and the structure is not tending to an equilibrium state. Where the solution is not convergent, this will result in a non-convergence error in the analysis.

See User Defined Non-Linear Convergence Settings.