Structural Analysis

The program analyzes beams and columns using matrix stiffness analysis. The finite method employed is known as the direct stiffness method and can be found in engineering texts on beam theory and matrix methods in structural analysis.

Assumptions

The method assumes the following:

• It is a first-order method. Any second-order effects caused by the loads acting on the deflected member are assumed to be so small that they can be safely ignored
• Transverse sections which are plane before bending remain plain after bending
• It is an elastic method. The member experiences small strains and will return to its original shape when loads are removed.
• Vertical deflections are very small with respect to beam length 
• Beam supports fully resist all x, y and z translations even if that support is another beam, that is, the effect of flexible supports is not considered
• The member is initially straight and all longitudinal filaments bend in circular arcs.

These are all reasonable assumptions for wood beams and columns loaded by forces of realistic magnitude.

Model

Each span between supports or a free cantilever end is modelled as a separate element, and the support points at the ends of the elements are nodes.

The method utilizes the force-displacement relationship of each element expressed as a stiffness matrix relating unknown nodal displacements to known nodal forces. In the case of a cantilever beam, the unknown displacements at the supports are the nodal rotations, and at the free end of the cantilever element. they are rotation and vertical displacement.

Where there is more than one element, such as a multi-span beam, individual elements are mathematically assembled into a single stiffness matrix of equations representing the entire beam or column. The equations are then solved to determine the nodal displacements caused by externally applied loads.

Nodal Forces

From these nodal displacements it is possible to calculate the resulting bending moment, shear force and axial force at each node by plugging the displacements into the stiffness matrix equations for each element. Support reactions are found by summing up the forces contributed by each element sharing a common node.

Interior to Span Forces

Once the values at the support nodes are known, bending moments and shear forces along the span of an element are calculated using simple beam statics.

Interior to Span Deflections

Given the rotation at the nodes, the deflections along the span of an member are calculated using standard slope-deflection "shape" equations, to which are added the deflections that would occur for fixed (non-rotated) ends using analytic formulae for each load distribution.

Currently only bending deflection is considered, shear deflection is not implemented at this time.