The P-Delta analysis utilised in MasterFrame is a P-Δ analysis, which modifies the frame geometry under loading and re-analyses the model in the deflected position. The software carries out an iterative process, carrying out the analysis a number of times, modifying the frame geometry for each iteration, until either an equilibrium condition is reached, or, if equilibrium is not reached, a frame stability warning is given.
Two methods of P-delta analysis are available, the Newton Raphson method or the Geometric Stiffness method.
1. Geometric Stiffness method
In the Geometric Stiffness the method, the bending stiffness of the elements in any model are modified to account for the axial force present in the member. Compressive forces will reduce the bending stiffness, while tensile axial forces will increase the element stiffness. In the case of a member which is loaded to its Euler buckling compressive load, the bending stiffness will be reduced to zero.
In the Geometric stiffness method, only two iterations are carried out, the first to calculate the axial force under a linear static analysis, the second iteration to account for the modified member stiffnesses in bending.
Since the geometric method only uses two iterations, it is less computationally demanding that the Newton-Raphson method, so, for larger models, it requires less time to analyse than would be required for the Newton Raphson method. However, with only two iterations, the method takes less account of the geometric deformation of the structure under load. Thus, the Geometric Stiffness method is less suitable for the analysis of structures where the deformation of the structure is significant. As a result, the Geometric Stiffness is less generally applicable than the Newton Raphson method.
2. Newton-Raphson method
The Newton-Raphson non-linear iterative method carries out a series of analyses, modifying the stiffness matrix of the structure to account for the deformation of the structure from the previous analysis. At each stage of the analysis, the out-of-balance forces due to the difference in the external and internal model forces is calculated. The process is continued until the out of balance in the external and internal forces is within a prescribed tolerance, within a specified number of iterations.
The process terminates when either (a) The analysis converges to a solution where the out of balance forces are within tolerance, or, (b) The analysis diverges and a solution is not possible.
The Newton Raphson method uses the lateral translations of the nodes within a model to assess the deformation of the model geometry in each iteration. This means the internal deformation of the members themselves is not necessarily assessed as part of the analysis, unless the member contains intermediate nodes. Intermediate nodes can be added to a member by splitting the member (see Modify Geometry>Split Member for details). Alternatively, analytic nodes can be added to a member. Where internal nodes are added to a member, then the deformation of the member is accounted for, to some degree, within the analysis.
In general, the member deformations are of less significance than the nodal deformations of the model. In terms of the member design, in the concrete design additional moments are added as the design stage to account for the internal deformations of the members, while in the steel and timber design, effective lengths and member buckling as accounted for at the design stage.
The Newton-Raphson method would be the most generally applicable type of P-delta analysis.
See User Defined Non-Linear Convergence Settings.