Governing equations of plane elasticity are examined to define suitable approximate theories. Each dependent variable in the problem is considered as a series expansion in Legendre polynomials; attention is focused on establishment of a logical approach to truncation of the series. Important variables for approximate theories of any order are established from energy considerations, and the desired approximate theories are established by direct reduction of the field equations and also from an energy viewpoint. A new “classical” beam theory is developed capable of treating displacement boundary conditions on lateral surfaces. Higher-order approximate theories are studied to make certain comparisons with exact solutions; the results of these comparisons indicate that the new method yields approximate theories which may be more accurate than previous theories with similar levels of approximation.

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