Structural Analysis

The program analyzes beams and columns using matrix stiffness analysis. The 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 modeled as a separate element, and the support points at the ends of the elements, or a cantilever end, are nodes. On each element, this method utilizes the force-displacement relationship expressed as a stiffness matrix relating unknown nodal displacements to known nodal forces.

The unknown displacements at beam supports are the rotations at the supports; the vertical displacement is zero. At the free end of the cantilever element. the displacements are rotation and vertical displacement. The local nodal forces at each end of the span are a point load and a moment at that are the equivalent to the effect of all the loads on the span.

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 global nodal displacements.

Global Nodal Forces

From these nodal displacements it is possible to calculate the force and bending moment at each end of the element 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.

Refer also to Shear Deflection.