# Methods of calculation (Frame Analysis)

This description of the methods of calculation is not meant to show the process of the program in detail, but serves to give enough information on how the calculation result has been attained.

# Analysis

Frame Analysis executes the frame calculations with the finite element method where the displacement method is used to determine displacements and stresses for the whole structure by using stiffness-relations for the individual members.

The construction is divided into members and joints. The joints are those points where the members are connected. The joint conception can however be perceived in a more strict mathematical meaning, as a discreet point in the pattern that describes the structure.

The members of the construction are first being analysed and general relations between forces and displacements of the joints are being set up, these are the
stiffness relations for the members. The demands of continuity and equilibrium in the joints can then be formulated with matrix-expressions and the result will be a connection between joint forces and joint displacements for the whole construction, which is the stiffness-relation of the system. The connection can generally be written in matrix-form as:

• K a = f

Where

• K is a global stiffness-matrix
• a is joint displacements
• f contains the joint forces.

The course of the calculation can be summarised as:

• Define the problem, divide the structure into beam-members and joints and initiate support as boundary conditions.
• Form member equations, i.e. connections between forces and displacements for separate members.
• Assemble; this means that the member equations are being put into the equilibrium relations for the joints, which results in an equation system for the whole structure.
• The equation system is being solved with consideration to current boundary conditions. At this the joint displacements and the reaction forces will be known.
• Member equations and the now known displacements determine the member forces.

Consideration is taken to the alteration in the forces of the members, caused by deformation of the structure, when using the second order theory.

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