Residual stresses caused by growth stresses within a stem with radially varying spiral grain angle - two numerical solution approaches: 1) finite element method and 2) transfer matrix method

Abstract

Two numerical methods, the Finite Element (FE) method and the Transfer Matrix (TM) method, have been applied to derive a solution for residual stresses in a stem. The source of stress is the maturation strains in new wood cells, which give rise to growth stresses. The methods are generally applicable to the case of spiral grain with a grain angle varying with radius, which often is the case in trees. Also, the methods allow for the elastic constants to vary with the radius, which is important to model, for example, juvenile wood. The numerical solutions are checked against an analytical solution in the case of a constant grain angle, and are found to coincide. In addition, the two methods are shown to independently produce the same stress distribution for one pattern of grain angle variation. This shows that the methods are reliable, and they will be used for obtaining a wider range of solutions in a subsequent paper.