Journal of Applied Mechanics · 1963 · 72 citations · 0 references
EngineeringMechanical EngineeringSolid MechanicsStructural AnalysisStructural EngineeringBeam AnalysisBeam TheoryBending Of BeamsDeflection Of BeamsStress AnalysisStructural MechanicsBeam Deflection AnalysisBeam-bending AnalysisBendingBending Stress AnalysisBernoulli EquationStress Distribution In BeamsVariable Cross-section Beams
The study examines stresses and deflections of thin rectangular beams with variable depth in pure bending, noting that the Bernoulli‑Euler theory ambiguously defines the beam axis and offers several admissible choices. The authors adopt a series‑based solution to facilitate comparison between the Bernoulli‑Euler approximation and the exact plane‑stress solution. The solution is expressed as a series whose first term matches the strength‑of‑materials result, with subsequent terms providing corrections to the theory. The Bernoulli‑Euler approximation is shown to be highly accurate for thin beams when the beam edge ordinates and their spanwise derivatives satisfy specific conditions.
The stresses and deflections of thin rectangular beams of arbitrary variable depth, in pure bending, according to the theory of plane stress, are considered. They are obtained in the form of series; the first term of each series is identical with the strength-of-materials solution and the others represent the necessary correction to that theory. This form of the solution is chosen because of its convenience in the study of the relationship between the Bernoulli-Euler and the exact solution. The former is found to be quite accurate for thin beams and, when certain conditions are satisfied by the ordinates (and their spanwise derivatives) of the upper and lower edges of the beam. The Bernoulli-Euler theory is ambiguous in prescribing the position of the axis of a beam of variable cross section; admissible choices for the axis are presented.