MasterFrame Analysis

MasterFrame allows the user to construct a representation of the structure to be analysed by the use of MasterFrame physical members, area loads including gravity and wind loads, member and nodal loads and also to model foundation and other restraint to movements of the structure using nodal supports. The MasterFrame physical model is an idealised representation of the actual structure.

Under lying the MasterFrame physical model is the analytic model, comprised of analytic nodes and analytic members. The analytic model is a further idealised abstraction of the structure. The underlying analytic model is what is used to set up the system of equations which are solved to form the analysis of the structure.

The section properties of the MasterFrame physical members are converted into member stiffnesses for the analytic members of the analytic model. The stiffnesses are initially determined relative to the local member axes. Each analytic member in the model will have its own local stiffness properties relating to the axial and bending stiffness of the member. All analytic member stiffnesses can be expressed in matrix form. Each analytic member stiffness matrix is then transformed into global coordinates. At this stage, the transformed member stiffness matrices are compiled into a global stiffness matrix which represents the stiffness matrix of the full structure in the global coordinates.

The system of equations which represent the structure to be analysed can be written, in matrix form, as: -

F = K d

Where F and d are column vectors representing the forces and moments acting at the analytic nodes of the model and the displacements and rotations of the analytic nodes, all in global coordinates.

The stiffness matrix K has the following properties: -

1.The stiffness matrix is square. The order of the matrix is equal to the number of degrees of freedom in the analytical model.

2.The stiffness matrix is symmetric

3.The matrix K is positive-definite

4.The structural matrix K is non-singular.

The first 3 points above relate to properties of the matrix which allow certain operations to be carried out on the matrix which form part of the process of finding a solution to the system of equations.

Where a matrix is singular, this means that it has either no solutions, or infinitely many solutions. In both cases, a single solution does not exist. This would indicate that the structure is unstable. Hence a singular matrix is indicative of a structural instability in the physical model.

Following manipulation of the set system of equations, where the stiffness matrix K is non-singular, the system of equations is solved to give the displacements and rotations of each analytic node in the model. This represents the displacement of the structure relative to the global system of axes. The global displacements are then converted into local displacements for each analytic member of the model, from which the end reactions and end moments of the analytic members is calculated. At this stage, the member loading, in the form of linearly varying loads and point loads on the analytic members, are used to calculate the bending moment and shear force distribution on the member. From the bending moments and shear force diagrams, combined with the end displacements and rotations of the nodes, the deflected shape of the member is calculated.

The above briefly describes the analysis method which is carried out for a 1^st order linear elastic analysis of an analytic model for a single load case. For model which included multiple load cases, the analysis process is carried out for each load case, with the results of each analysis being saved. While superposition of load cases can be used for 1^st order linearly elastic analysis, this cannot be used for non-linear analysis and so the software does not use superposition.

To accommodate non-linearity in the analysis, MasterFrame uses an iterative process to solve the system of equations. In this case, the analysis of the analytic model in any load case is carried out a number of times, with a modification of the stiffness matrix being carried out after each stage of the analysis. A convergence criterion is used to determine when the change in the solution is small enough to consider the solution to be complete, or, alternatively, to determine when the solution is divergent and no solution is possible. A modification of this process can accommodate a 2^nd order P-delta analysis, which can account for the geometric modification of the structure under loading.